When the graph of the function y = f(x), defined on the real numbers, is reflected in the y-axis and then translated by 2 units in the negative x-direction, the result is the graph of the function y = g(x).
When the graph of the same function y = f(x) is translated by 2 units in the negative x-direction and then reflected in the y-axis, the result is the graph of the function y = h(x).
Which one of the following conditions on y = f(x) is necessary and sufficient for the functions g(x) and h(x) to be identical?
- Af(x) = f(x + 2) for all x
- Bf(x) = f(x + 4) for all x
- Cf(x) = f(x + 8) for all x
- Df(x) = f(−x) for all x
- Ef(x) = f(2 − x) for all x
- Ff(x) = f(4 − x) for all x
- Gf(x) = f(8 − x) for all x
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- Af(x) = f(x + 2) for all x
- Bf(x) = f(x + 4) for all x
- Cf(x) = f(x + 8) for all x
- Df(x) = f(−x) for all x
- Ef(x) = f(2 − x) for all x
- Ff(x) = f(4 − x) for all x
- Gf(x) = f(8 − x) for all x
Apply the transformations to the argument one at a time. Reflecting in the y-axis gives f(−x); translating that 2 to the left replaces x by x + 2, so g(x) = f(−x−2). The other order gives f(x+2) first, and reflecting replaces x by −x, so h(x) = f(2 − x). Setting them equal, f(−x−2) = f(2−x) for all x; writing u = −x − 2 turns the right-hand argument into u + 4, so the condition is f(u) = f(u+4) for all u — periodicity with period 4 (or a factor of it).