Advertisements
Advertisements
Question
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx + 1", if" x <= 5),(3x - 5", if" x > 5):}` at x = 5
Advertisements
Solution
f(x) = `{(kx + 1", if" x <= 5),(3x - 5", if" x > 5):}`
If f(x) is continuous at x = 5, this implies:
f(5) = `lim_(x -> 5^+)` f(x) = `lim_(x -> 5^-)` f(x)
⇒ k(5) + 1 = 3(5) − 5
⇒ 5k + 1 = 15 − 5
⇒ 5k + 1 = 10
⇒ 5k = 10 − 1
⇒ 5k = 9
⇒ k = `9/5`
That is, for the quantity k = `9/5` this function is continuous at x = 5.
APPEARS IN
RELATED QUESTIONS
A function f (x) is defined as
f (x) = x + a, x < 0
= x, 0 ≤x ≤ 1
= b- x, x ≥1
is continuous in its domain.
Find a + b.
For what value of λ is the function defined by f(x) = `{(λ(x^2 - 2x)", if" x <= 0),(4x+ 1", if" x > 0):}` continuous at x = 0? What about continuity at x = 1?
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx^2", if" x<= 2),(3", if" x > 2):}` at x = 2
Find the values of a and b such that the function defined by f(x) = `{(5", if" x <= 2),(ax +b", if" 2 < x < 10),(21", if" x >= 10):}` is a continuous function.
Show that the function defined by f(x) = cos (x2) is a continuous function.
Show that the function defined by f(x) = |cos x| is a continuous function.
Examine the continuity of the function
\[f\left( x \right) = \left\{ \begin{array}{l}3x - 2, & x \leq 0 \\ x + 1 , & x > 0\end{array}at x = 0 \right.\]
Also sketch the graph of this function.
Find the values of a so that the function
If \[f\left( x \right) = \begin{cases}\frac{\cos^2 x - \sin^2 x - 1}{\sqrt{x^2 + 1} - 1}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, find k.
Extend the definition of the following by continuity
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}\frac{1 - \cos 2kx}{x^2}, \text{ if } & x \neq 0 \\ 8 , \text{ if } & x = 0\end{cases}\] at x = 0
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}(x - 1)\tan\frac{\pi x}{2}, \text{ if } & x \neq 1 \\ k , if & x = 1\end{cases}\] at x = 1at x = 1
If \[f\left( x \right) = \begin{cases}\frac{x^2}{2}, & \text{ if } 0 \leq x \leq 1 \\ 2 x^2 - 3x + \frac{3}{2}, & \text P{ \text{ if } } 1 < x \leq 2\end{cases}\]. Show that f is continuous at x = 1.
The function \[f\left( x \right) = \begin{cases}x^2 /a , & \text{ if } 0 \leq x < 1 \\ a , & \text{ if } 1 \leq x < \sqrt{2} \\ \frac{2 b^2 - 4b}{x^2}, & \text{ if } \sqrt{2} \leq x < \infty\end{cases}\] is continuous on (0, ∞), then find the most suitable values of a and b.
If \[f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x}\]
for x ≠ π/4, find the value which can be assigned to f(x) at x = π/4 so that the function f(x) becomes continuous every where in [0, π/2].
Discuss the continuity of the following functions:
(i) f(x) = sin x + cos x
(ii) f(x) = sin x − cos x
(iii) f(x) = sin x cos x
Show that f (x) = | cos x | is a continuous function.
What happens to a function f (x) at x = a, if
If f (x) = (x + 1)cot x be continuous at x = 0, then f (0) is equal to
Find the values of a and b so that the function
If \[f\left( x \right) = \begin{cases}\frac{\left| x + 2 \right|}{\tan^{- 1} \left( x + 2 \right)} & , x \neq - 2 \\ 2 & , x = - 2\end{cases}\] then f (x) is
The function f (x) = x − [x], where [⋅] denotes the greatest integer function is
If \[f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\]
then at x = 0, f (x) is
If f(x) = 2x and g(x) = `x^2/2 + 1`, then which of the following can be a discontinuous function ______.
The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.
The function f(x) = `"e"^|x|` is ______.
If f.g is continuous at x = a, then f and g are separately continuous at x = a.
Let `"f" ("x") = ("In" (1 + "ax") - "In" (1 - "bx"))/"x", "x" ne 0` If f (x) is continuous at x = 0, then f(0) = ____________.
The point(s), at which the function f given by f(x) = `{("x"/|"x"|"," "x" < 0),(-1"," "x" ≥ 0):}` is continuous, is/are:
The value of f(0) for the function `f(x) = 1/x[log(1 + x) - log(1 - x)]` to be continuous at x = 0 should be
If `f`: R → {0, 1} is a continuous surjection map then `f^(-1) (0) ∩ f^(-1) (1)` is:
The function f(x) = x2 – sin x + 5 is continuous at x =
For what value of `k` the following function is continuous at the indicated point
`f(x) = {{:(kx^2",", if x ≤ 2),(3",", if x > 2):}` at x = 2
