Question

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Explanation:

Step1: Check y-intercept

The graph intersects the y - axis at \(x = 0\). Let's find the value of each function at \(x=0\).

• For \(f(x)=5 - 2^{x}\): \(f(0)=5 - 2^{0}=5 - 1 = 4\)
• For \(f(x)=2^{1 - x}\): \(f(0)=2^{1-0}=2^{1}=2\)
• For \(f(x)=2^{x}-5\): \(f(0)=2^{0}-5 = 1 - 5=-4\)
• For \(f(x)=-2^{x - 1}\): \(f(0)=-2^{-1}=-\frac{1}{2}\)

The y - intercept of the graph is 4, so we can eliminate the functions with \(f(0)
eq4\). So we eliminate \(f(x)=2^{1 - x}\), \(f(x)=2^{x}-5\) and \(f(x)=-2^{x - 1}\).

Step2: Check the behavior as \(x\) increases

The graph is decreasing as \(x\) increases. Let's analyze \(f(x)=5 - 2^{x}\). The function \(y = 2^{x}\) is an increasing exponential function. So \(y=-2^{x}\) is a decreasing exponential function, and \(y = 5-2^{x}\) is also a decreasing function (since we subtract an increasing function from 5). As \(x\) increases, \(2^{x}\) increases, so \(5 - 2^{x}\) decreases, which matches the behavior of the graph.

Answer:

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