Question

f(x)

x	f(x)
-2	0
-1	1
0	2
1	3
2	4

g(x)
(graph of g(x))
which statement is true when comparing the table for f(x) to the graph of g(x)?
a both functions are exponential.
b both functions are linear.
c f(x) is a linear function, and g(x) is an exponential function.

Explanation:

Step1: Analyze \( f(x) \)

Check the table for \( f(x) \). The change in \( x \) is \( 1 \) (from -2 to -1, -1 to 0, etc.). The change in \( f(x) \) is \( 1 \) (0 to 1, 1 to 2, etc.). So, the slope \( m=\frac{\Delta f(x)}{\Delta x}=\frac{1}{1} = 1\), constant. Thus, \( f(x) \) is linear (form \( f(x)=x + 2 \), since at \( x = -2 \), \( f(-2)=0=-2 + 2 \), etc.).

Step2: Analyze \( g(x) \)

The graph of \( g(x) \) is a curve that decreases rapidly then levels off, characteristic of an exponential decay function (e.g., \( g(x)=a^x \) with \( 0 < a < 1 \)). It is not linear (no constant slope).

Step3: Evaluate Options

• Option A: \( f(x) \) is linear, not exponential. Eliminate.
• Option B: \( g(x) \) is exponential, not linear. Eliminate.
• Option C: \( f(x) \) is linear, \( g(x) \) is exponential. Correct.

Answer:

C. \( f(x) \) is a linear function, and \( g(x) \) is an exponential function.