Chapter 4.1, Problem 1E

The function f(x) = 5^x is an exponential function with base _____; f(−2) = _____, f(0) = _____, f(2) = _____, and f(6) = _____.

To determine

To Evaluate: The base of an exponential function $f\left(x\right)=5^x$ and the value for the function $f\left(-2\right),f\left(0\right),f\left(2\right)\text{and}f\left(6\right)$.

Answer to Problem 1E

The base of an exponential function $f\left(x\right)=5^x$ is 5 and the value of the function $f\left(-2\right)$ is $\frac{1}{25}$, $f\left(0\right)$ is 1, $f\left(2\right)$ is 25 and $f\left(6\right)$ is 15,625.

Explanation of Solution

The exponential expression of a function is, $f\left(x\right)=a^x$.

Where,

•  a is base of the function.
• x is any real number.

The given equation of the exponential function is, $f\left(x\right)=5^x$.

So, the base of the above function is 5.

Hence, the function $f\left(x\right)=5^x$ is an exponential function with base 5.

The given equation of the exponential function is, $f\left(x\right)=5^x$.

Substitute $-2$ for x in the above equation,

$\begin{array}{c}f\left(-2\right)=5^{-2}\\ f\left(-2\right)=\frac{1}{5^2}\\ f\left(-2\right)=\frac{1}{25}\end{array}$

Hence, the value of the exponential function $f\left(-2\right)$ is $\frac{1}{25}$.

Substitute 0 for x in the above equation,

$\begin{array}{c}f\left(0\right)=5^0\\ f\left(0\right)=1\end{array}$

Hence, the value of the exponential function $f\left(0\right)$ is 1.

Substitute 2 for x in the above equation,

$\begin{array}{c}f\left(2\right)=5^2\\ f\left(2\right)=25\end{array}$

Hence, the value of the exponential function $f\left(2\right)$ is 25.

Substitute 6 for x in the above equation,

$\begin{array}{c}f\left(6\right)=5^6\\ f\left(6\right)=15625\end{array}$

Hence, the value of the exponential function $f\left(6\right)$ is 15,625.