Question

The function$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathbf{5}}^{\mathbf{x}}$is an exponential function with base ____;

role="math" localid="1648559050478" $\mathit{f}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{=}\mathbf{\_}\mathbf{\_}\mathbf{\_}\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{\_}\mathbf{\_}\mathbf{\_}\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{=}\mathbf{\_}\mathbf{\_}\mathbf{\_}\mathbf{,}\mathbf{\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}}\mathit{f}\mathbf{\left(}\mathbf{6}\mathbf{\right)}\mathbf{=}\mathbf{\_}\mathbf{\_}\mathbf{\_}$

Short answer

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

$f(-2)=\frac{1}{25},f\left(0\right)=1,f\left(2\right)=25,\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}f\left(6\right)=15625$

Step-by-step solution

Step-1 – Given 

The given function is $f\left(x\right)=5^x$.

Step-2 – To determine

We have to find the base of the function, and the values of$f(-2),f\left(0\right),f\left(2\right),\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}f\left(6\right).$

Step-3 – Calculation

We compare $f\left(x\right)=5^x$with a standard exponential form $f\left(x\right)=a^x$.

Here, a = base = 5.

To find the values of $f(-2),f\left(0\right),f\left(2\right),\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}f\left(6\right)$we’ll plug $x=-2,0,2\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}6$in respectively $f\left(x\right)$.

So,

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

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

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

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

Hence, the base is 5 and $f(-2)=\frac{1}{25},f\left(0\right)=1,f\left(2\right)=25,\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}f\left(6\right)=15625$.