Question

match the exponential function with its graph.
(a) ( f(x) = 2^x )
--select--
(b) ( f(x) = 2^{-x} )
--select--
(c) ( f(x) = -2^x )
--select--
(d) ( f(x) = -2^{-x} )
--select--

Explanation:

Step1: Analyze \(f(x)=2^x\)

Exponential growth: \(f(0)=1\), increases as \(x\) increases. Matches Graph III.

Step2: Analyze \(f(x)=2^{-x}\)

Exponential decay: \(f(0)=1\), decreases as \(x\) increases. Matches Graph I.

Step3: Analyze \(f(x)=-2^x\)

Reflected over x-axis: \(f(0)=-1\), decreases as \(x\) increases. Matches Graph II.

Step4: Analyze \(f(x)=-2^{-x}\)

Reflected over x-axis: \(f(0)=-1\), increases as \(x\) increases. Matches Graph IV.

Answer:

(a) III
(b) I
(c) II
(d) IV