Question

the function g(x) is shown on the graph. if f(x) = -3x - 5, which statement is true when comparing f(x) to g(x).
a f(x) is increasing and g(x) is decreasing
b f(x) is decreasing and g(x) is increasing
c both functions are increasing
d both functions are decreasing

Explanation:

Step1: Analyze \( f(x) \)

The function \( f(x) = -3x - 5 \) is a linear function. The slope of a linear function \( y = mx + b \) determines if it's increasing or decreasing. Here, the slope \( m = -3 \), which is negative. So, as \( x \) increases, \( f(x) \) decreases (since slope is negative, the function is decreasing).

Step2: Analyze \( g(x) \)

Looking at the graph of \( g(x) \), we observe the trend of the curve. For the part of the graph shown, as \( x \) increases (moving from left to right), the \( y \)-values of \( g(x) \) are increasing (the curve is rising, especially the exponential - like part on the right). So \( g(x) \) is an increasing function.

Step3: Compare \( f(x) \) and \( g(x) \)

From Step 1, \( f(x) \) is decreasing. From Step 2, \( g(x) \) is increasing. So the statement " \( f(x) \) is decreasing and \( g(x) \) is increasing" is true.

Answer:

B. \( f(x) \) is decreasing and \( g(x) \) is increasing