Question

x | f(x)
-2 | 12.5
-1 | 2.5
0 | 0.5
1 | 0.1
2 | 0.02
which exponential function is represented by the table?
○ f(x) = 0.2(0.5^x)
○ f(x) = 0.5(5^x)
○ f(x) = 0.5(0.2^x)
○ f(x) = 0.2(0.2^x)

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: Find the initial value (\(a\))

When \( x = 0 \), \( f(0) = 0.5 \). For \( f(x) = a(b^x) \), when \( x = 0 \), \( b^0 = 1 \), so \( f(0) = a(1) = a \). Thus, \( a = 0.5 \).

Step3: Find the base (\(b\))

Use another point, e.g., \( x = 1 \), \( f(1) = 0.1 \). Substitute \( a = 0.5 \) and \( x = 1 \) into \( f(x) = a(b^x) \): \( 0.1 = 0.5(b^1) \). Solve for \( b \): \( b=\frac{0.1}{0.5}=0.2 \)? Wait, no, wait. Wait, let's check \( x = -1 \). When \( x=-1 \), \( f(-1)=2.5 \). Using \( f(x)=0.5(b^x) \), \( 2.5 = 0.5(b^{-1}) \). Then \( b^{-1}=\frac{2.5}{0.5}=5 \), so \( b=\frac{1}{5}=0.2 \)? Wait, no, wait. Wait, maybe I made a mistake. Wait, let's check the option \( f(x)=0.5(5^x) \). When \( x=-1 \), \( 0.5(5^{-1})=0.5\times\frac{1}{5}=0.1 \), which is not 2.5. Wait, no, the table has \( x=-1 \), \( f(-1)=2.5 \). Let's check option \( f(x)=0.5(5^x) \): when \( x=-1 \), \( 0.5\times5^{-1}=0.5\times\frac{1}{5}=0.1 \), not 2.5. Wait, maybe I messed up. Wait, let's check the first option: \( f(x)=0.2(0.5^x) \). When \( x=-2 \), \( 0.2(0.5^{-2})=0.2\times4=0.8 \), not 12.5. Second option: \( f(x)=0.5(5^x) \). When \( x=-2 \), \( 0.5(5^{-2})=0.5\times\frac{1}{25}=0.02 \), not 12.5. Third option: \( f(x)=0.5(0.2^x) \). When \( x=-2 \), \( 0.5(0.2^{-2})=0.5\times25=12.5 \), which matches. When \( x=-1 \), \( 0.5(0.2^{-1})=0.5\times5=2.5 \), matches. When \( x=0 \), \( 0.5(0.2^0)=0.5\times1=0.5 \), matches. When \( x=1 \), \( 0.5(0.2^1)=0.5\times0.2=0.1 \), matches. When \( x=2 \), \( 0.5(0.2^2)=0.5\times0.04=0.02 \), matches. So the correct function is \( f(x)=0.5(0.2^x) \).

Answer:

\( f(x) = 0.5(0.2^x) \) (the third option: \( f(x) = 0.5(0.2^x) \))