Question

which exponential function is represented by the graph?
options:

• ( f(x) = 2(3^x) )
• ( f(x) = 3(3^x) )
• ( f(x) = 3(2^x) )
• ( f(x) = 2(2^x) )

the graph has points (-1, 1.5), (0, 3), (1, 6) plotted on a coordinate plane with x from -5 to 5 and y from 1 to 9.

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( f(x) = a(b^x) \), where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base.

Step2: Use the point (0, 3)

When \( x = 0 \), \( f(0) = 3 \). Substitute \( x = 0 \) into each option:

• For \( f(x)=2(3^x) \): \( f(0)=2(3^0)=2(1)=2

eq3 \)

• For \( f(x)=3(3^x) \): \( f(0)=3(3^0)=3(1)=3 \), but we'll check another point.
• For \( f(x)=3(2^x) \): \( f(0)=3(2^0)=3(1)=3 \)
• For \( f(x)=2(2^x) \): \( f(0)=2(2^0)=2(1)=2

eq3 \)

So we can eliminate \( f(x)=2(3^x) \) and \( f(x)=2(2^x) \).

Step3: Use the point (1, 6)

Now check \( x = 1 \) for the remaining options (\( f(x)=3(3^x) \) and \( f(x)=3(2^x) \)):

• For \( f(x)=3(3^x) \): \( f(1)=3(3^1)=3(3)=9

eq6 \)

• For \( f(x)=3(2^x) \): \( f(1)=3(2^1)=3(2)=6 \), which matches the point (1, 6).

We can also check the point (-1, 1.5) with \( f(x)=3(2^x) \): \( f(-1)=3(2^{-1})=3(\frac{1}{2}) = 1.5 \), which matches.

Answer:

C. \( f(x) = 3(2^x) \)