Question #f6c3c

1 Answer
Jan 26, 2018

Shove in the value of g(3) into the function f(x) to get that (f @ g)(3) = -22.

Explanation:

First, let's see what (f @ g)(x) means.

(f @ g)(x), or perhaps simply f @ g(x), is shorthand for f(g(x)). That means shoving in a value for x, solving for g(x) by shoving in that value of x, then shoving in that solved value into the function f, hence f(g(x)).

f @ g(3) is then simply f(g(3)). We're already given the value of x, which is 3, and we're told to solve this function composition: first solving for g(3), then solving for the function f having shoved in the value of g(3).

Here, we have f(x) = -2x + 8 and g(x) = 5x. Since an equation, any equation, tells us that both sides are equal (and can be replaced by each other), why don't we replace g(x) with 5x?

Doing that, f(g(x)) becomes f(5x), so f(g(x)) = f(5x).

Since f(x) = -2x + 8 for any input x (which means we might as well say f(n) = -2n + 8 and it would still be the same), solving for f(5x) would simply be shoving in that 5x as the input:

f(5x) = -2(5x) + 8

Simplify:

f(5x) = -10x + 8

Summarizing what we've done so far, we have:

(f @ g)(x) = f @ g(x) = f(g(x)) = f(5x) = -10x + 8

Now, we just need to solve for x = 3:

(f @ g)(3) = -10(3) + 8

= -30 + 8 = -22

Therefore, (f @ g)(3) = -22.