Question

the graph of the function f is shown below. which statement about the function is true? o f is an even function. o 0≤f(x)≤1 for all real numbers. o f is differentiable everywhere. o f is continuous everywhere.

Explanation:

Step1: Recall continuity concept

A function is continuous if there are no breaks, jumps or holes in its graph. Looking at the graph of \(y = f(x)\), we can see that the graph has no such irregularities.

Step2: Analyze differentiability

A function is not differentiable at sharp - corners. The graph has sharp - corners, so it is not differentiable everywhere.

Step3: Check for even - function property

An even function has \(f(x)=f( - x)\), i.e., the graph is symmetric about the y - axis. The given graph is not symmetric about the y - axis.

Step4: Check range

We cannot say that \(0\leq f(x)\leq1\) for all real numbers just from the given part of the graph.

Answer:

f is continuous everywhere.