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Let F: R → R Be Defined as F(X) = 10x + 7. Find the Function G: R → R Such that G O F = F O G = 1r. - Mathematics

Let fR → be defined as f(x) = 10x + 7. Find the function gR → R such that g o f = f o = 1R.

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Solution

It is given that fR → R is defined as f(x) = 10x + 7.

One-one:

Let f(x) = f(y), where xy ∈R.

⇒ 10x + 7 = 10y + 7

⇒ x = y

∴ is a one-one function.

Onto:

For ∈ R, let y = 10x + 7.

`=> x=  (y -7)/10 in R `

Therefore, for any ∈ R, there exists `x = (y-7)/10 in R`

such that

`f(x) = f((y -7)/10) = 10((y -7)/10) + 7 = y - 7  + 7 = y`

∴ is onto.

Therefore, is one-one and onto.

Thus, f is an invertible function.

Let us define gR → R as `g(y) = (y -7)/10`

Now, we have:

`gof(x) = g(f(x)) = g(10x + 7) = ((10x + 7) - 7)/10 = 10x/10 = 10`

And

`fog(y) = f(g(y)) = f((y -7)/10) = 10 ((y-7)/10) + 7 = y - 7 + 7 =   y`

Hence, the required function gR → R is defined as `g(y) = (y - 7)/10`

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APPEARS IN

NCERT Class 12 Maths
Chapter 1 Relations and Functions
Q 1 | Page 29
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