Question

If f is an even function, determine whether g is even, odd, or neither. Explain.

(a)

g(x) = −f(x)

even

odd

neither

Explain.

The graph is a reflection in the x-axis.

The graph is a reflection in the y-axis.

The graph is a vertical translation of f.

The graph is a horizontal translation of f.

The graph is symmetric about the origin.

(b)

g(x) = f(−x)

even

odd

neither

Explain.

The graph is a reflection in the x-axis.

The graph is a reflection in the y-axis.

The graph is a vertical translation of f.

The graph is a horizontal translation of f.

The graph is symmetric about the origin.

(c)

g(x) = f(x) − 3

even

odd

neither

Explain.

The graph is a reflection in the x-axis.

The graph is a reflection in the y-axis.

The graph is a vertical translation of f.

The graph is a horizontal translation of f.

The graph is symmetric about the origin.

(d)

g(x) = f(x − 3)

even

odd

neither

The graph is a reflection in the x-axis.

The graph is a reflection in the y-axis.

The graph is a vertical translation of f.

The graph is a horizontal translation of f.

The graph is symmetric about the origin.

Step 1

Given: f(x) is an even function. Hence by definition, f(-x) = f(x). We will use this result in all the subparts of the question solved below.

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