Question

x | f(x)
-2 | 0.2
-1 | 0.4
0 | 0.8
1 | 1.6
2 | 3.2
which exponential function is represented by the table?
○ f(x) = 2(2^x)
○ f(x) = 0.8(0.8^x)
○ f(x) = 2(0.8^x)
○ f(x) = 0.8(2^x)

Explanation:

Step1: Recall the 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 (the common ratio between consecutive terms).

Step2: Find the initial value \( a \)

When \( x = 0 \), \( f(0) = a(b^0)=a(1)=a \). From the table, when \( x = 0 \), \( f(0)=0.8 \), so \( a = 0.8 \).

Step3: Find the base \( b \)

Let's check the ratio between consecutive \( f(x) \) values. For \( x=-1 \) to \( x = 0 \): \( \frac{f(0)}{f(-1)}=\frac{0.8}{0.4} = 2 \). For \( x = 0 \) to \( x = 1 \): \( \frac{f(1)}{f(0)}=\frac{1.6}{0.8}=2 \). So the base \( b = 2 \).

Step4: Form the exponential function

Using \( a = 0.8 \) and \( b = 2 \), the function is \( f(x)=0.8(2^x) \).

Answer:

\( f(x) = 0.8(2^x) \) (the last option: \( f(x) = 0.8(2^x) \))