Question

homework.derivita.com
f(x) = 2^x - 5
f(x) = -2^{x - 1}
b)
graph of a curve on a grid with x-axis from -6 to 6 and y-axis from -6 to 6, passing through (0,2) and decreasing
f(x) = 5 - 2^x
f(x) = 2^{1 - x}
f(x) = 2^x - 5
f(x) = -2^{x - 1}

Explanation:

Step1: Analyze the graph's key features

The graph is a decreasing exponential curve (since it falls from left to right) and passes through the point \((0, 2)\) (when \(x = 0\), \(y = 2\)). Let's check each function at \(x = 0\):

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

eq2\).

• For \(f(x)=2^{1 - x}\): At \(x = 0\), \(f(0)=2^{1 - 0}=2^{1}=2\). This matches the \(y\)-intercept.
• For \(f(x)=2^{x}-5\): At \(x = 0\), \(f(0)=2^{0}-5 = 1 - 5=-4

eq2\).

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

eq2\).

Step2: Confirm the function's behavior

The function \(f(x)=2^{1 - x}\) can be rewritten as \(f(x)=2\times2^{-x}=\frac{2}{2^{x}}\), which is a decreasing exponential function (since the exponent of \(2\) is \(-x\), so as \(x\) increases, \(2^{-x}\) decreases, making the whole function decrease), which matches the graph's decreasing nature. The other functions either don't match the \(y\)-intercept or the behavior.

Answer:

\(f(x) = 2^{1 - x}\) (the option with \(f(x)=2^{1 - x}\))