Question

which of the following is the graph of $f(x) = \frac{1}{5} \cdot 10^{x + 1} - 3$

Explanation:

Step1: Analyze the function type

The function \( f(x)=\frac{1}{5}\cdot10^{x + 1}-3 \) is an exponential function. Since the base \( 10>1 \), the function is an increasing exponential function (as exponential functions with base \( a>1 \) are increasing). So we can eliminate the graphs that are decreasing (the first, second, and fourth graphs as they have a decreasing curve).

Step2: Check the y - intercept

To find the y - intercept, set \( x = 0 \):
\( f(0)=\frac{1}{5}\cdot10^{0 + 1}-3=\frac{1}{5}\times10 - 3=2 - 3=- 1 \)
Now, check the remaining graph (the orange - colored one) to see if when \( x = 0 \), the y - value is - 1. Also, as \( x\to-\infty \), \( 10^{x + 1}\to0 \), so \( f(x)\to\frac{1}{5}\times0-3=-3 \), which means the horizontal asymptote is \( y=-3 \). As \( x\) increases, the function increases because the base of the exponential is greater than 1. The orange graph shows an increasing exponential curve, matches the y - intercept at \( x = 0,y=-1 \) and the horizontal asymptote \( y = - 3 \) (as \( x\to-\infty \), the function approaches \( y=-3 \)).

Answer:

The graph with the orange background (the third graph from the top) is the graph of \( f(x)=\frac{1}{5}\cdot10^{x + 1}-3 \)