Question

select the correct answer. consider the graph of the function $f(x)=2^{x}$. which statement describes a key feature of function $g$ if $g(x)=2f(x)$? a. horizontal asymptote of $y = 2$ b. horizontal asymptote of $y=-2$ c. y - intercept at $(0,2)$ d. y - intercept at $(2,0)$

Explanation:

Step1: Find the y - intercept of \(f(x)\)

For \(f(x)=2^{x}\), when \(x = 0\), \(f(0)=2^{0}=1\), so the y - intercept of \(f(x)\) is \((0,1)\).

Step2: Find the y - intercept of \(g(x)\)

Given \(g(x)=2f(x)\), when \(x = 0\), \(g(0)=2f(0)\). Since \(f(0) = 1\), then \(g(0)=2\times1 = 2\). So the y - intercept of \(g(x)\) is \((0,2)\).

Step3: Analyze the horizontal asymptote

The function \(f(x)=2^{x}\) has a horizontal asymptote \(y = 0\). When we have \(g(x)=2f(x)=2\times2^{x}=2^{x + 1}\), it still has a horizontal asymptote \(y = 0\).

Answer:

C. y - intercept at \((0,2)\)