Question

Question was provided via image upload.

Explanation:

Step1: Analyze the graph's behavior

The graph in part (a) is an exponential growth curve (increasing as \( x \) increases) with a vertical intercept (when \( x = 0 \)) at \( y=-4 \). Let's check each function at \( x = 0 \):

• For \( f(x)=5 - 2^{x} \): \( f(0)=5 - 2^{0}=5 - 1 = 4 \). Not -4.
• For \( f(x)=2^{1 - x} \): \( f(0)=2^{1 - 0}=2^{1}=2 \). Not -4.
• For \( f(x)=2^{x}-5 \): \( f(0)=2^{0}-5 = 1 - 5=-4 \). Matches the y - intercept. Also, as \( x \) increases, \( 2^{x} \) increases, so \( f(x)=2^{x}-5 \) is an increasing exponential function, which matches the graph.

Step2: Confirm the function type

The graph is an exponential growth (since it's increasing) shifted down. The function \( f(x)=2^{x}-5 \) is a standard exponential function \( 2^{x} \) shifted down by 5 units, which has the correct shape (increasing, y - intercept at -4) as the given graph.

Answer:

\( f(x)=2^{x}-5 \)