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Q. 120

Expert-verified
Found in: Page 287

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# If $f\left(x\right)={a}^{x}$, show that $f\left(-x\right)=\frac{1}{f\left(x\right)}$.

To prove $f\left(-x\right)=\frac{1}{f\left(x\right)}$, first find $f\left(-x\right)$ and apply the laws of exponents.

See the step by step solution

## Step 1. Given information.

Consider the given question,$f\left(x\right)={a}^{x}\phantom{\rule{0ex}{0ex}}f\left(-x\right)=\frac{1}{f\left(x\right)}$

Then,

$f\left(-x\right)={a}^{-x}$

## Step 2. Use the law of exponents.

Using laws of exponents, ${m}^{-n}=\frac{1}{{m}^{n}}$,

$f\left(-x\right)=\frac{1}{{a}^{x}}\phantom{\rule{0ex}{0ex}}f\left(-x\right)=\frac{1}{f\left(x\right)}$

Therefore, $LHS=RHS$.

Hence, proved.