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If [tex]$g(x) = -x^2 - 5x + 1$[/tex] and [tex]$h(x) = (x + 3)^2 - 11$[/tex], find [tex][tex]$g(h(-5))$[/tex][/tex].

A. [tex]-13[/tex]
B. [tex]5[/tex]
C. [tex]-149[/tex]

Sagot :

GinnyAnswer
Sure, let's solve this problem step-by-step together.

We have two functions defined as follows:
1. [tex]\( g(x) = -x^2 - 5x + 1 \)[/tex]
2. [tex]\( h(x) = (x + 3)^2 - 11 \)[/tex]

To find [tex]\( g(h(-5)) \)[/tex], we first need to evaluate [tex]\( h(x) \)[/tex] at [tex]\( x = -5 \)[/tex].

Step 1: Evaluate [tex]\( h(-5) \)[/tex]

[tex]\[ h(x) = (x + 3)^2 - 11 \][/tex]
Replace [tex]\( x \)[/tex] with [tex]\(-5\)[/tex]:

[tex]\[ h(-5) = (-5 + 3)^2 - 11 \][/tex]
[tex]\[ h(-5) = (-2)^2 - 11 \][/tex]
[tex]\[ h(-5) = 4 - 11 \][/tex]
[tex]\[ h(-5) = -7 \][/tex]

So, [tex]\( h(-5) = -7 \)[/tex].

Step 2: Evaluate [tex]\( g(h(-5)) \)[/tex], which is essentially evaluating [tex]\( g(-7) \)[/tex]

We now have [tex]\( h(-5) = -7 \)[/tex]. So, next we need to find [tex]\( g(-7) \)[/tex]:

[tex]\[ g(x) = -x^2 - 5x + 1 \][/tex]
Replace [tex]\( x \)[/tex] with [tex]\(-7\)[/tex]:

[tex]\[ g(-7) = -(-7)^2 - 5(-7) + 1 \][/tex]
[tex]\[ g(-7) = -49 + 35 + 1 \][/tex]
[tex]\[ g(-7) = -49 + 36 \][/tex]
[tex]\[ g(-7) = -13 \][/tex]

Therefore, [tex]\( g(h(-5)) = g(-7) = -13 \)[/tex].

So the final answer is [tex]\(-13\)[/tex].
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