To find [tex]\( g^{-1}(-7) \)[/tex], we need to determine the value of [tex]\( x \)[/tex] such that [tex]\( g(x) = -7 \)[/tex]. We will evaluate each piece of the function [tex]\( g(x) \)[/tex] to see where [tex]\( g(x) = -7 \)[/tex] holds true.
### Case 1: [tex]\( g(x) = 5x^2 \)[/tex] when [tex]\( x \leq -3 \)[/tex]
For this case, we set up the equation:
[tex]\[ 5x^2 = -7 \][/tex]
However, [tex]\( 5x^2 \)[/tex] is always non-negative for all real [tex]\( x \)[/tex]. Since [tex]\(-7\)[/tex] is negative, this equation has no real solutions. Therefore, there are no values of [tex]\( x \)[/tex] in this interval that satisfy [tex]\( g(x) = -7 \)[/tex].
### Case 2: [tex]\( g(x) = 21 + x \)[/tex] when [tex]\( -3 < x \leq 10 \)[/tex]
For this case, we set up the equation:
[tex]\[ 21 + x = -7 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = -7 - 21
\][/tex]
[tex]\[
x = -28
\][/tex]
To check if this solution is valid, we need to ensure that [tex]\(-28\)[/tex] lies within the interval [tex]\(-3 < x \leq 10\)[/tex]. Clearly, [tex]\(-28\)[/tex] is not within this interval. Therefore, there are no values of [tex]\( x \)[/tex] in this interval that satisfy [tex]\( g(x) = -7 \)[/tex].
### Case 3: [tex]\( g(x) = 2 - \sqrt{x} \)[/tex] when [tex]\( x > 10 \)[/tex]
For this case, we set up the equation:
[tex]\[ 2 - \sqrt{x} = -7 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
2 + 7 = \sqrt{x}
\][/tex]
[tex]\[
9 = \sqrt{x}
\][/tex]
Squaring both sides:
[tex]\[
81 = x
\][/tex]
We need to check if [tex]\( 81 \)[/tex] lies within the interval [tex]\( x > 10 \)[/tex]. Clearly, [tex]\( 81 \)[/tex] is greater than [tex]\( 10 \)[/tex], so this solution is valid.
Hence, the value of [tex]\( x \)[/tex] that satisfies [tex]\( g(x) = -7 \)[/tex] is:
[tex]\[ g^{-1}(-7) = 81 \][/tex]
