Question

use the description and table to graph the function, and determine the domain and range of f(x). represent the domain and range with inequality notation, interval notation, or set - builder notation. explain your reasoning.
the table has two rows. the first row (lets assume its the x - values) has: - 3, $\frac{1}{125}$, - 2, $\frac{1}{25}$, - 1, $\frac{1}{5}$, 0, 1, 1, 5, 2, 25, 3, 125 (the formatting of the table in the original image might be a bit unclear, but this is the text - related part of the table).

Explanation:

Step1: Identify the function

First, observe the table values:
When $x=-3$, $f(x)=\frac{1}{125}=5^{-3}$
When $x=-2$, $f(x)=\frac{1}{25}=5^{-2}$
When $x=-1$, $f(x)=\frac{1}{5}=5^{-1}$
When $x=0$, $f(x)=1=5^{0}$
When $x=1$, $f(x)=5=5^{1}$
When $x=2$, $f(x)=25=5^{2}$
When $x=3$, $f(x)=125=5^{3}$
So the function is $f(x)=5^x$.

Step2: Determine the domain

For exponential functions of the form $a^x$ ($a>0,a
eq1$), there are no restrictions on $x$; any real number can be input.
Domain in interval notation: $(-\infty, \infty)$
In inequality notation: $-\infty < x < \infty$

Step3: Determine the range

Since $5^x$ is always positive for any real $x$, and as $x\to-\infty$, $5^x\to0$ (never reaching 0), and as $x\to\infty$, $5^x\to\infty$.
Range in interval notation: $(0, \infty)$
In inequality notation: $0 < f(x) < \infty$

Answer:

Function: $f(x)=5^x$
Domain (interval notation): $(-\infty, \infty)$; (inequality notation): $-\infty < x < \infty$
Range (interval notation): $(0, \infty)$; (inequality notation): $0 < f(x) < \infty$

To graph this function: plot the points from the table, then draw a smooth curve that approaches the x-axis (y=0) as $x$ decreases to negative infinity, and increases rapidly as $x$ increases to positive infinity, passing through the point $(0,1)$.