Chapter 5: Q. 77 (page 258)

If $f$ is an odd function and $g$ is an even function, show that the composite functions $f \circ g$ and $g \circ f$ are both even.

Short Answer

Expert verified

The, $f \circ g$ and $g \circ f$ both are even function.

Step by step solution

Step 1. Given Information

Given that $f$ is odd function and $g$ is even function.

Step 2. Solution

A function $f$ is said to be odd if $f (- x) = - f (x) \forall x .$

A function $g$ is said to be even if role="math" localid="1646315555532" $g (- x) = g (x) \forall x .$

Since

$f \circ g (x) = f (g (x))$

Now,

$f \circ g (- x) = f (g (- x)) f \circ g (- x) = f (g (x)) {g i s e v e n f u n c t i o n .} f \circ g (- x) = f \circ g (x) \forall x$

So, $f \circ g$ is an even function.

For $g \circ f (x) = g (f (x)) .$

$g \circ f (- x) = g (f (- x)) g \circ f (- x) = g (- f (x)) {f i s o d d f u n c t i o n} g \circ f (- x) = g (f (x)) {g i s e v e n f u n c t i o n} g \circ f (- x) = g \circ f (x) \forall x$

So $g \circ f$ is an even function.

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If $f$ is an odd function and $g$ is an even function, show that the composite functions $f \circ g$ and $g \circ f$ are both even.

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