Q. 120 - Precalculus Enhanced with Graphing Utilities

Question

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

Step-by-Step Solution

Answer

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

Step 1. Given information.

Consider the given question,
$$\begin{gathered} f\left(x\right)=a^x \\ f\left(-x\right)=\frac{1}{f\left(x\right)} \end{gathered}$$

Then,

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

Step 2. Use the law of exponents.

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

$$\begin{gathered} f\left(-x\right)=\frac{1}{a^x} \\ f\left(-x\right)=\frac{1}{f\left(x\right)} \end{gathered}$$

Therefore, $LHS=RHS$.

Hence, proved.