Question

The graph of the function \(f\) is to be transformed as described. Find the function for the transformed graph. \(f(x)=x \sin x\); stretched horizontally by a factor of 2

Short answer

The function \(f(x) = x\sin x\) stretched horizontally by a factor of 2 results in the transformed function: \(g(x) = \frac{x}{2} \sin\left(\frac{x}{2}\right)\).

Step-by-step solution

Understand the horizontal stretch transformation

A horizontal stretch by a factor of 2 means that all horizontal distances in the graph are multiplied by 2. Mathematically, this can be achieved by replacing every instance of \(x\) with \(\frac{x}{2}\) in the function definition.

Apply the transformation to the given function

Now, we will apply the horizontal stretch to the function \(f(x) = x \sin x\). Replace \(x\) with \(\frac{x}{2}\) in the given function: \(g(x) = \left(\frac{x}{2}\right) \sin\left(\frac{x}{2}\right)\)

Simplify the transformed function if necessary

In this case, there are no additional simplifications that can be made, and the transformed function is: \(g(x) = \frac{x}{2} \sin\left(\frac{x}{2}\right)\)

Final Answer

The function \(f(x) = x\sin x\) stretched horizontally by a factor of 2 results in the transformed function: \(g(x) = \frac{x}{2} \sin\left(\frac{x}{2}\right)\)