Question

the graph of f(x) is shown. which graph represents g(x) = f(2x)?

Explanation:

Step1: Recall the horizontal compression rule

For a function \(y = f(kx)\) where \(k>1\), the graph of \(y = f(x)\) is horizontally compressed by a factor of \(\frac{1}{k}\). Here \(k = 2\), so the graph of \(y=f(x)\) is horizontally compressed by a factor of \(\frac{1}{2}\).

Step2: Analyze the key - points

Let's assume a key - point \((x_0,y_0)\) on the graph of \(y = f(x)\). For the function \(y=f(2x)\), when \(y = y_0\), \(2x=x_0\) (since \(y_0=f(x_0)\) and \(y_0 = f(2x)\)), then \(x=\frac{x_0}{2}\). This means that for each \(y\) - value, the \(x\) - value on the graph of \(y = f(2x)\) is half of the \(x\) - value on the graph of \(y = f(x)\).

Answer:

The graph that is a horizontal compression (by a factor of \(\frac{1}{2}\)) of the given graph of \(y = f(x)\) (the second graph from the left among the four options) represents \(g(x)=f(2x)\).