Sure! To find [tex]\( (g \circ f)(x) \)[/tex], also written as [tex]\( g(f(x)) \)[/tex], we need to compose the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex]. This means we will substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]. Here are the given functions:
[tex]\[ f(x) = 5 - x \][/tex]
[tex]\[ g(x) = x^2 + 3x - 2 \][/tex]
Let's find [tex]\( g(f(x)) \)[/tex] step-by-step.
1. Substitute [tex]\( f(x) = 5 - x \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(5 - x) \][/tex]
2. Calculate [tex]\( g(5 - x) \)[/tex]:
[tex]\[ g(5 - x) = (5 - x)^2 + 3(5 - x) - 2 \][/tex]
3. Expand [tex]\( (5 - x)^2 \)[/tex]:
[tex]\[ (5 - x)^2 = 25 - 10x + x^2 \][/tex]
4. Substitute this back into [tex]\( g(5 - x) \)[/tex]:
[tex]\[ g(5 - x) = 25 - 10x + x^2 + 3(5 - x) - 2 \][/tex]
[tex]\[ g(5 - x) = 25 - 10x + x^2 + 15 - 3x - 2 \][/tex]
5. Combine the like terms:
[tex]\[ g(5 - x) = x^2 - 13x + 38 \][/tex]
So, [tex]\( (g \circ f)(x) = x^2 - 13x + 38 \)[/tex].
Next, we will evaluate this composition at the specified points: [tex]\( x = 0 \)[/tex], [tex]\( x = 1 \)[/tex], and [tex]\( x = -1 \)[/tex].
Evaluating [tex]\( g(f(0)) \)[/tex]:
[tex]\[ g(f(0)) = (0)^2 - 13(0) + 38 = 38 \][/tex]
Evaluating [tex]\( g(f(1)) \)[/tex]:
[tex]\[ g(f(1)) = (1)^2 - 13(1) + 38 = 1 - 13 + 38 = 26 \][/tex]
Evaluating [tex]\( g(f(-1)) \)[/tex]:
[tex]\[ g(f(-1)) = (-1)^2 - 13(-1) + 38 = 1 + 13 + 38 = 52 \][/tex]
So, the values of [tex]\( (g \circ f)(x) \)[/tex] are:
[tex]\[
(g \circ f)(0) = 38, \quad (g \circ f)(1) = 26, \quad (g \circ f)(-1) = 52
\][/tex]
