If f(x) = x + 1 and g(x) = x – 1,(a) f(9(x)) =(b) g(f(x)) =(c) Thus g(x) is called anfunction of f(x)

Answered: If f(x) = x +1 and g(x) = x – 1, (a)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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If f(x) = x + 1 and g(x) = x – 1,
(a) f(9(x)) =
(b) g(f(x)) =
(c) Thus g(x) is called an
function of f(x)

Expert Solution

a) answer

Given that

$f (x) = x + 1$ and $g (x) = x - 1$

$f (g (x))$ is calculated by applying g function first and f function later to the obtained expression.

Firstly, $g (x) = x - 1$ , consider this as y.

Then, $f (g (x)) = f (y)$

$f (x) = x + 1$ similarly when x is replaced with y, $f (y) = y + 1$

So,

$\begin{array}{rcl} f (g (x)) & = & f (y) \\ = & y + 1 \\ = & (x - 1) + 1 \\ = & x \end{array}$

Therefore, $f (g (x)) = x$

b) answer

Given functions are $f (x) = x + 1$ and $g (x) = x - 1$

$g (f (x))$ is calculated by applying f function first and then g function later to the obtained expression.

Firstly, the function f is given as $f (x) = x + 1$ , consider this as y.

That is, let $y = x + 1$

Then, $g (f (x)) = g (y)$

Now, apply function g: $g (x) = x - 1$ similarly when x is replaced with y, $g (y) = y - 1$

So, the value of $g (f (x))$ is:

$\begin{array}{rcl} g (f (x)) & = & g (y) \\ = & y - 1 \\ = & (x + 1) - 1 \\ = & x \end{array}$

Therefore, $g (f (x)) = x$

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If f(x) = x +1 and g(x) = x – 1, (a) f(g(x)) = (b) 9(f(x)) = (c) Thus g(x) is called an function of f(x)