Given the definitions of f, of, xf(x) and g, of, xg(x) below, find the value of f, of, g, of, minus, 2, .f(g(−2)).

Question

Given the definitions of f, of, xf(x) and g, of, xg(x) below, find the value of f, of, g, of, minus, 2, .f(g(−2)).
f, of, x, equals, 3, x, squared, minus, 7, x, plus, 15
f(x)=
3x 
2
 −7x+15
g, of, x, equals, x, plus, 7
g(x)=
x+7

GPT-4o mini · Answer

To solve the expression $ f(g(-2)) $ step-by-step, we first need to calculate $ g(-2) $ using the definition of $ g(x) $. 1. **Calculate $ g(-2) $**: $$ g(x) = x + 7 $$ Substituting $-2$ into $g(x)$: $$ g(-2) = -2 + 7 = 5 $$ 2. **Now, calculate $ f(g(-2)) = f(5) $**: Using the definition of $ f(x) $: $$ f(x) = 3x^2 - 7x + 15 $$ Substituting $5$ into $f(x)$: $$ f(5) = 3(5^2) - 7(5) + 15 $$ First, compute $5^2$: $$ 5^2 = 25 $$ Now calculate $f(5)$: $$ f(5) = 3(25) - 35 + 15 = 75 - 35 + 15 $$ Combining these: $$ 75 - 35 = 40 $$ $$ 40 + 15 = 55 $$ So, $f(5) = 55$. 3. **Finally, we need to determine $f(g(-2))$ and we have found it equals $55$**. The original question seems to ask for $f(g(-2))$ without additional operations such as subtraction or further composition, so: $$ \text{Value of } f(g(-2)) = 55 $$ In conclusion, the answer is: $$ \boxed{55} $$