Balbharati solutions for मैथमेटिक्स एण्ड स्टैटिस्टिक्स २ (आर्ट्स एण्ड सायन्स) [अंग्रेजी] कक्षा ११ महाराष्ट्र स्टेट बोर्ड chapter 6 - Functions [Latest edition]

Check if the following relation is a function.

Check if the following relation is a function.

Check if the following relation is a function.

Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {−1, 0, 1, 2, 3}? Justify.

{(1, 0), (3, 3), (2, −1), (4, 1), (2, 2)}

Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {−1, 0, 1, 2, 3}? Justify.

{(1, 2), (2, −1), (3, 1), (4, 3)}

Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {−1, 0, 1, 2, 3}? Justify.

{(1, 3), (4, 1), (2, 2)}

Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {−1, 0, 1, 2, 3}? Justify.

{(1, 1), (2, 1), (3, 1), (4, 1)}

Check if the relation given by the equation represents y as function of x:

2x + 3y = 12

Check if the relation given by the equation represents y as function of x:

x + y² = 9

Check if the relation given by the equation represents y as function of x:

x² − y = 25

Check if the relation given by the equation represents y as function of x:

2y + 10 = 0

Check if the relation given by the equation represents y as function of x:

3x − 6 = 21

If f(m) = m² − 3m + 1, find f(0)

If f(m) = m² − 3m + 1, find f(−3)

If f(m) = m² − 3m + 1, find `f(1/2)`

If f(m) = m² − 3m + 1, find f(x + 1)

If f(m) = m² − 3m + 1, find f(− x)

If f(m) = m² − 3m + 1, find `(("f"(2 + "h") - "f"(2))/"h"), "h" ≠ 0`

Find x, if g(x) = 0 where g(x) = `(5x - 6)/7`

Find x, if g(x) = 0 where g(x) = `(18 -2x^2)/7`

Find x, if g(x) = 0 where g(x) = 6x² + x − 2

Find x, if g(x) = 0 where g(x) = x³ − 2x² − 5x + 6

Find x, if f(x) = g(x) where f(x) = x⁴ + 2x², g(x) = 11x²

Find x, if f(x) = g(x) where f(x) = `sqrt(x) - 3`, g(x) = 5 – x

If f(x) = `("a" - x)/("b" - x)`, f(2) is undefined, and f(3) = 5, find a and b

Find the domain and range of the following function.

f(x) = 7x² + 4x − 1

Find the domain and range of the following function.

g(x) = `(x + 4)/(x - 2)`

Find the domain and range of the follwoing function.

h(x) = `sqrt(x + 5)/(5 + x)`

Find the domain and range of the following function.

f(x) = `root(3)(x + 1)`

Find the domain and range of the following function.

f(x) = `sqrt((x - 2)(5 - x)`

Find the domain and range of the following function.

f(x) = `sqrt((x - 3)/(7 - x))`

Find the domain and range of the following function.

f(x) = `sqrt(16 - x^2)`

Express the area A of a square as a function of its side s

Express the area A of a square as a function of its perimeter P

Express the area A of circle as a function of its radius r

Express the area A of circle as a function of its diameter d

Express the area A of circle as a function of its circumference C.

An open box is made from a square of cardboard of 30 cms side, by cutting squares of length x centimeters from each corner and folding the sides up. Express the volume of the box as a function of x. Also find its domain

Let f be a subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z? Justify?

Check the injectivity and surjectivity of the following function.

f : N → N given by f(x) = x² 

Check the injectivity and surjectivity of the following function.

f : Z → Z given by f(x) = x² 

Check the injectivity and surjectivity of the following function.

f : R → R given by f(x) = x² 

Check the injectivity and surjectivity of the following function.

f : N → N given by f(x) = x³

Check the injectivity and surjectivity of the following function.

f : R → R given by f(x) = x³ 

Show that if f : A → B and g : B → C are one-one, then g ° f is also one-one

Show that if f : A → B and g : B → C are onto, then g ° f is also onto

lf f(x) = 3(4^x+1), find f(– 3)

Express the following exponential equation in logarithmic form

2⁵ = 32

Express the following exponential equation in logarithmic form

54° = 1

Express the following exponential equation in logarithmic form

23¹ = 23

Express the following exponential equation in logarithmic form

`9^(3/2)` = 27

Express the following exponential equation in logarithmic form

3^–4 = `1/81`

Express the following exponential equation in logarithmic form

10⁻² = 0.01

Express the following exponential equation in logarithmic form

e² = 7.3890

Express the following exponential equation in logarithmic form

`"e"^(1/2)` = 1.6487

Express the following exponential equation in logarithmic form

e^–x = 6

Express the following logarithmic equation in exponential form

log₂ 64 = 6

Express the following logarithmic equation in exponential form

`log_5  1/25` = – 2

Express the following logarithmic equation in exponential form

log₁₀ (0.001) = −3

Express the following logarithmic equation in exponential form

`log_(1/2) (8)` = – 3

Express the following logarithmic equation in exponential form

ln 1 = 0

Express the following logarithmic equation in exponential form

ln e = 1

Express the following logarithmic equation in exponential form

In `1/2` = – 0.693

Find the domain of f(x) = ln (x − 5)

Find the domain of f(x) = log₁₀ (x² − 5x + 6)

Write the following expression as sum or difference of logarithm

`log ("pq"/"rs")`

Write the following expression as sum or difference of logarithm

`log (sqrt(x) root(3)(y))`

Write the following expression as sum or difference of logarithm

In `(("a"^3 ("a" - 2)^2)/sqrt("b"^2 + 5))`

Write the following expression as sum or difference of logarithm

In `[(root(3)(x - 2)(2x + 1)^4)/((x + 4)sqrt(2x + 4))]^2`

Write the following expression as a single logarithm.

5 log x + 7 log y − log z

Write the following expression as a single logarithm.

`1/3 log (x - 1) + 1/2 log (x)`

Write the following expression as a single logarithm.

ln (x + 2) + ln (x − 2) − 3 ln (x + 5)

Given that log 2 = a and log 3 = b, write `log sqrt(96)` in terms of a and b

Prove that `"b"^(log_"b""a"` = a

Prove that log_b^m a = `1/"m" log_"b""a"`

Prove that a^log_cb = b^log_ca

If f(x) = ax² − bx + 6 and f(2) = 3 and f(4) = 30, find a and b

Solve for x.

log2 + log(x + 3) – log(3x – 5) = log3

Solve for x.

2 log₁₀ x = `1 + log_10 (x + 11/10)`

Solve for x.

log₂ x + log₄ x + log₁₆ x = `21/4`

Solve for x.

x + log₁₀ (1 + 2^x) = x log₁₀ 5 + log₁₀ 6

If `log((x + y)/3) = 1/2 log x + 1/2 logy`, show that `x/y + y/x` = 7

If `log(( x - y)/4) = logsqrt(x) + log sqrt(y)`, show that (x + y)² = 20xy 

If x = log_a (bc), y = log_b (ca), z = log_c (ab) then prove that `1/(1 + x) + 1/(1 + y) + 1/(1 + z)` = 1

If f(x) = 3x + 5, g(x) = 6x − 1, then find (f + g) (x)

If f(x) = 3x + 5, g(x) = 6x − 1, then find (f − g) (2)

If f(x) = 3x + 5, g(x) = 6x − 1, then find (fg) (3)

If f(x) = 3x + 5, g(x) = 6x − 1, then find `("f"/"g") (x)` and its domain

Let f : {2, 4, 5} → {2, 3, 6} and g : {2, 3, 6} → {2, 4} be given by f = {(2, 3), (4, 6), (5, 2)} and g = {(2, 4), (3, 4), (6, 2)}. Write down g ° f

If f(x) = 2x² + 3, g (x) = 5x − 2, then find f ° g

If f(x) = 2x² + 3, g(x) = 5x − 2, then find g ° f

If f(x) = 2x² + 3, g(x) = 5x − 2, then find f ° f

If f(x) = 2x² + 3, g (x) = 5x − 2, then find g ° g

Verify that f and g are inverse functions of each other, where f(x) = `(x - 7)/4`, g(x) = 4x + 7

Verify that f and g are inverse functions of each other, where f(x) = x³ + 4, g(x) = `root(3)(x - 4)`

Verify that f and g are inverse functions of each other, where f(x) = `(x + 3)/(x - 2)`, g(x) = `(2x + 3)/(x - 1)`

Check if the following function has an inverse function. If yes, find the inverse function.

f(x) = 5x²

Check if the following function has an inverse function. If yes, find the inverse function.

f(x) = 8

Check if the following function has an inverse function. If yes, find the inverse function.

f(x) = `(6x - 7)/3`

Check if the following function has an inverse function. If yes, find the inverse function.

f(x) = `sqrt(4x + 5)`

Check if the following function has an inverse function. If yes, find the inverse function.

f(x) = 9x³ + 8

Check if the following function has an inverse function. If yes, find the inverse function.

f(x) = `{(x + 7, x < 0),(8 - x, x ≥ 0):}`

If f(x) = `{(x^2 + 3, x ≤ 2),(5x + 7, x > 2):}`, then find f(3)

If f(x) = `{(x^2 + 3, x ≤ 2),(5x + 7, x > 2):}`, then find f(2)

If f(x) = `{(x^2 + 3, x ≤ 2),(5x + 7, x > 2):}`, then find f(0)

If f(x) = `{(4x - 2",", x ≤ -3),(5",", -3 < x < 3),(x^2",", x ≥ 3):}`, then find f(– 4)

If f(x) = `{(4x - 2",", x ≤ -3),(5",", -3 < x < 3),(x^2",", x ≥ 3):}`, then find f(– 3)

If f(x) = `{(4x - 2",", x ≤ -3),(5",", -3 < x < 3),(x^2",", x ≥ 3):}`, then find f(1)

If f(x) = `{(4x - 2",", x ≤ -3),(5",", -3 < x < 3),(x^2",", x ≥ 3):}`, then find f(5)

If f(x) = 2|x| + 3x, then find f(2)

If f(x) = 2|x| + 3x, then find f(– 5)

If f(x) = 4[x] − 3, where [x] is greatest integer function of x, then find f(7.2)

If f(x) = 4[x] − 3, where [x] is greatest integer function of x, then find f(0.5)

If f(x) = 4[x] − 3, where [x] is greatest integer function of x, then find `"f"(- 5/2)`

If f(x) = 4[x] − 3, where [x] is greatest integer function of x, then find f(2π), where π = 3.14

If f(x) = 2{x} + 5x, where {x} is fractional part function of x, then find f(– 1)

If f(x) = 2{x} + 5x, where {x} is fractional part function of x, then find `"f"(1/4)`

If f(x) = 2{x} + 5x, where {x} is fractional part function of x, then find f(– 1.2)

If f(x) = 2{x} + 5x, where {x} is fractional part function of x, then find f(– 6)

Solve the following for x, where |x| is modulus function, [x] is greatest integer function, [x] is a fractional part function.

|x + 4| ≥ 5

Solve the following for x, where |x| is modulus function, [x] is greatest integer function, [x] is a fractional part function.

|x − 4| + |x − 2| = 3

Solve the following for x, where |x| is modulus function, [x] is greatest integer function, [x] is a fractional part function.

x² + 7 |x| + 12 = 0

Solve the following for x, where |x| is modulus function, [x] is greatest integer function, [x] is a fractional part function.

|x| ≤ 3

Solve the following for x, where |x| is modulus function, [x] is greatest integer function, [x] is a fractional part function.

2|x| = 5

Solve the following for x, where |x| is modulus function, [x] is greatest integer function, [x] is a fractional part function.

[x + [x + [x]]] = 9

Solve the following for x, where |x| is modulus function, [x] is greatest integer function, [x] is a fractional part function.

{x} > 4

Solve the following for x, where |x| is modulus function, [x] is greatest integer function, [x] is a fractional part function.

{x} = 0

Solve the following for x, where |x| is modulus function, [x] is greatest integer function, [x] is a fractional part function.

{x} = 0.5

Solve the following for x, where |x| is modulus function, [x] is greatest integer function, [x] is a fractional part function.

2{x} = x + [x]

Select the correct answer from given alternatives.

If log (5x – 9) – log (x + 3) = log 2 then x = ...............

Select the correct answer from given alternatives.

If log₁₀(log₁₀(log₁₀x)) = 0 then x =

Select the correct answer from given alternatives.

Find x, if 2log₂ x = 4

The equation log_x² 16 + log_2x 64 = 3 has,

Select the correct answer from given alternatives.

If f(x) =`1/(1 - x)`, then f{f[f(x)]} is

Select the correct answer from given alternatives.

If f : R → R is defined by f(x) = x³ then f^–1 (8) is equal to :

Select the correct answer from given alternatives.

Let the function f be defined by f(x) = `(2x + 1)/(1 - 3x)` then f^–1 (x) is ______.

Select the correct answer from given alternatives

If f(x) = 2x² + bx + c and f(0) = 3 and f(2) = 1, then f(1) is equal to

Select the correct answer from given alternatives

The domain of `1/([x] - x)` where [x] is greatest integer function is

Select the correct answer from given alternative.

The domain and range of f(x) = 2 − |x − 5| is

Answer the following:

Identify the following relation is the function? If it is a function determine its domain and range.

{(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

Answer the following:

Identify the following relation is the function? If it is a function determine its domain and range.

{(0, 0), (1, 1), (1, –1), (4, 2), (4, –2), (9, 3), (9, –3), (16, 4), (16, –4)}

Answer the following:

Identify the following relation is the function? If it is a function determine its domain and range

{(12, 1), (3, 1), (5, 2)}

Answer the following:

Find whether the following function is one-one

f : R → R defined by f(x) = x² + 5

Answer the following:

Find whether the following function is one-one

f : R − {3} → R defined by f(x) = `(5x + 7)/(x - 3)` for x ∈ R − {3}

Answer the following:

Find whether the following function is onto or not.

f : Z → Z defined by f(x) = 6x – 7 for all x ∈ Z

Answer the following:

Find whether the following function is onto or not.

f : R → R defined by f(x) = x² + 3 for all x ∈ R

Answer the following:

Let f: R → R be a function defined by f(x) = 5x³ – 8 for all x ∈ R, show that f is one-one and onto. Hence find f ^–1 

Answer the following:

A function f : R → R defined by f(x) = `(3x)/5 + 2`, x ∈ R. Show that f is one-one and onto. Hence find f^–1

Answer the following:

A function f is defined as f(x) = 4x + 5, for – 4 ≤ x < 0. Find the values of f(–1), f(–2), f(0), if they exist

A function f is defined as : f(x) = 5 – x for 0 ≤ x ≤ 4. Find the value of x such that f(x) = 3

Answer the following:

A function f is defined as : f(x) = 5 – x for 0 ≤ x ≤ 4. Find the value of x such that f(x) = 5

Answer the following:

If f(x) = 3x⁴ – 5x² + 7 find f(x – 1)

Answer the following:

If f(x) = 3x + a and f(1) = 7 find a and f(4)

Answer the following:

If f(x) = ax² + bx + 2 and f(1) = 3, f(4) = 42, find a and b

Answer the following:

Find composite of f and g:
f = {(1, 3), (2, 4), (3, 5), (4, 6)}
g = {(3, 6), (4, 8), (5, 10), (6, 12)}

Answer the following:

Find composite of f and g:
f = {(1, 1), (2, 4), (3, 4), (4, 3)}
g = {(1, 1), (3, 27), (4, 64)}

Answer the following:

Find f ° g and g ° f : f(x) = x² + 5, g(x) = x – 8

Answer the following:

Find f ° g and g ° f: f(x) = 3x – 2, g(x) = x²

Answer the following:

Find f ° g and g ° f: f(x) = 256x⁴, g(x) = `sqrt(x)`

Answer the following:

If f(x) = `(2x - 1)/(5x - 2), x ≠ 5/2` show that (f ° f) (x) = x

Answer the following:

If f(x) = `(x + 3)/(4x - 5)`, g(x) = `(3 + 5x)/(4x - 1)` then show that (f ° g) (x) = x

Answer the following:

Let f : R – {2} → R be defined by f(x) = `(x^2 - 4)/(x - 2)` and g : R → R be defined by g(x) = x + 2. Examine whether f = g or not

Answer the following:

Let f : R → R be given by f(x) = x + 5 for all x ∈ R. Draw its graph

Answer the following:

Let f : R → R be given by f(x) = x³ + 1 for all x ∈ R. Draw its graph

Answer the following:

For any base show that log (1 + 2 + 3) = log 1 + log 2 + log 3

Answer the following:

Find x, if x = 3^3log₃²  

Answer the following:

Show that, `log |sqrt(x^2 + 1) + x | + log | sqrt(x^2 + 1) - x|` = 0

Answer the following:

Show that, `log ("a"^2/"bc") + log ("b"^2/"ca") + log ("c"^2/"ab")` = 0

Answer the following:

Simplify, log (log x⁴) – log (log x)

Answer the following:

Simplify `log_10  28/45 - log_10  35/324 + log_10  325/432 - log_10  13/15`

Answer the following:

If `log (("a" + "b")/2) = 1/2(log"a" + log"b")`, then show that a = b

Answer the following:

If b² = ac. prove that, log a + log c = 2 log b

Answer the following:

Solve for x, log_x (8x – 3) – log_x 4 = 2

Answer the following:

If a² + b² = 7ab, show that, `log(("a" + "b")/3) = 1/2 log "a" + 1/2 log "b"`

Answer the following:

If `log ((x - y)/5) = 1/2 logx + 1/2 log y`, show that x² + y² = 27xy

Answer the following:

If f(x) = log(1 – x), 0 ≤ x < 1 show that `"f"(1/(1 + x))` = f(1 – x) – f(– x)

Answer the following:

Without using log tables, prove that `2/5 < log_10 3 < 1/2`

Answer the following:

Show that `7log (15/16) + 6log(8/3) + 5log (2/5) + log(32/25)` = log 3

Answer the following:

Solve : `sqrt(log_2 x^4) + 4log_4 sqrt(2/x)` = 2

Answer the following:

Find value of `(3 + log_10 343)/(2 + 1/2 log_10 (49/4) + 1/2 log_10 (1/25)`

Answer the following:

If `log"a"/(x + y - 2z) = log"b"/(y + z - 2x) = log"c"/(z + x - 2y)`, show that abc = 1

Answer the following:

Show that, log_y x³ . log_z y⁴ . log_x z⁵ = 60

Answer the following:

If `log_2"a"/4 = log_2"b"/6 = log_2"c"/(3"k")` and a³b²c = 1 find the value of k

Answer the following:

If a² = b³ = c⁴ = d⁵, show that log_a bcd = `47/30`

Answer the following:

Solve the following for x, where |x| is modulus function, [x] is greatest interger function, {x} is a fractional part function

1 < |x − 1| < 4

Answer the following:

Solve the following for x, where |x| is modulus function, [x] is greatest interger function, {x} is a fractional part function

|x² − x − 6| = x + 2

Answer the following:

Solve the following for x, where |x| is modulus function, [x] is greatest interger function, {x} is a fractional part function

|x² − 9| + |x² − 4| = 5

Answer the following:

Solve the following for x, where |x| is modulus function, [x] is greatest interger function, {x} is a fractional part function

−2 < [x] ≤ 7

Answer the following:

Solve the following for x, where |x| is modulus function, [x] is greatest interger function, {x} is a fractional part function

2[2x − 5] − 1 = 7

Answer the following:

Solve the following for x, where |x| is modulus function, [x] is greatest interger function, {x} is a fractional part function

[x²] − 5[x] + 6 = 0

Answer the following:

Solve the following for x, where |x| is modulus function, [x] is greatest interger function, {x} is a fractional part function

[x − 2] + [x + 2] + {x} = 0

Answer the following:

Solve the following for x, where |x| is modulus function, [x] is greatest interger function, {x} is a fractional part function

`[x/2] + [x/3] = (5x)/6`

Answer the following:

Find the domain of the following function.

f(x) = `(x^2 + 4x + 4)/(x^2 + x - 6)`

Answer the following:

Find the domain of the following function.

f(x) = `sqrt(x - 3) + 1/(log(5 - x))`

Answer the following:

Find the domain of the following function.

f(x) = `sqrt(1 - sqrt(1 - sqrt(1 - x^2)`

Answer the following:

Find the domain of the following function.

f(x) = x!

Answer the following:

Find the domain of the following function.

f(x) = ^5–xP_x–1

Answer the following:

Find the domain of the following function.

f(x) = `sqrt(x - x^2) + sqrt(5 - x)`

Find the domain of the following function.

f(x) = `sqrtlog(x^2 - 6x + 6)`

Answer the following:

Find the range of the following function.

f(x) = |x – 5|

Answer the following:

Find the range of the following function.

f(x) = `x/(9 + x^2)`

Answer the following:

Find the range of the following function.

f(x) = `1/(1 + sqrt(x))`

Answer the following:

Find the range of the following function.

f(x) = [x] – x

Answer the following:

Find the range of the following function.

f(x) = 1 + 2^x + 4^x 

Answer the following:

Find (f ° g) (x) and (g ° f) (x)

f(x) = e^x, g(x) = log x

Answer the following:

Find (f ° g) (x) and (g ° f) (x)

f(x) = `x/(x + 1)`, g(x) = `x/(1 - x)`

Answer the following:

Find f(x) if g(x) = `1 + sqrt(x)` and f[g(x)] = `3 + 2sqrt(x) + x`

Answer the following:

Find f(x) if g(x) = x² + x – 2 and (g ° f) (x) = 4x² – 10x + 4

Answer the following:

Find (f ° f) (x) if f(x) = `x/sqrt(1 + x^2)`

Answer the following:

Find (f ° f) (x) if f(x) = `(2x + 1)/(3x - 2)`