Question

Find functions f(x) and g(x) so the given function can be expressed as h(x) = f(g(x)). h(x) = 4/3 * x - 4/7

a) f(x) = 3x, g(x) = 4/7 * x
b) f(x) = 4/3 * x, g(x) = x - 4/7
c) f(x) = 3/4 * x, g(x) = x - 4/7
d) f(x) = 4/3 * x, g(x) = 7/4 * x

Answer 1

To express the function h(x) = 4/3 * x - 4/7 as a composition of two functions f(g(x)), we must find the correct pairing of f(x) and g(x). After checking the options, we find that option (b) f(x) = 4/3 * x, g(x) = x - 4/7 is the correct pair that leads to the original function when composed.

Step-by-step explanation:

The student is seeking to decompose the function h(x) = \frac{4}{3}x - \frac{4}{7} into two functions such that h(x) = f(g(x)). The correct answer is found by identifying a suitable inner function g(x) that when input into the outer function f(x) will result in the original function h(x). In this case, let's consider option (b) f(x) = \frac{4}{3}x, g(x) = x - \frac{4}{7}.

Now let's compose these functions to see if they give the desired original function:

f(g(x)) = f(x - \frac{4}{7})

= \frac{4}{3}(x - \frac{4}{7})

= \frac{4}{3}x - \frac{4}{3}\cdot\frac{4}{7}

= \frac{4}{3}x - \frac{16}{21}

= \frac{4}{3}x - \frac{4}{7}

Since composing functions f and g results in the original function h, the correct pair of functions is indeed option (b) f(x) = \frac{4}{3}x and g(x) = x - \frac{4}{7}.