Answer :
To determine if the functions [tex]\( f(x) = -2x + 8 \)[/tex] and [tex]\( g(x) = -\frac{1}{2}x - 8 \)[/tex] are inverses, we need to check the compositions [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex]. If both of these compositions simplify to [tex]\( x \)[/tex], then [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are indeed inverses.
### Step 1: Compute [tex]\( f(g(x)) \)[/tex]
First, substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f\left(-\frac{1}{2}x - 8\right) \][/tex]
Plug the expression for [tex]\( g(x) \)[/tex] into the function [tex]\( f \)[/tex]:
[tex]\[ f\left(-\frac{1}{2}x - 8\right) = -2\left(-\frac{1}{2}x - 8\right) + 8 \][/tex]
Next, simplify the expression inside the function [tex]\( f \)[/tex]:
[tex]\[ = -2 \left(-\frac{1}{2}x - 8\right) + 8 \][/tex]
Distribute [tex]\(-2\)[/tex] through the parentheses:
[tex]\[ = -2 \left(-\frac{1}{2}x \right) - 2(-8) + 8 \][/tex]
[tex]\[ = x + 16 + 8 \][/tex]
Combine like terms:
[tex]\[ = x + 24 \][/tex]
So,
[tex]\[ f(g(x)) = x + 24 \][/tex]
### Step 2: Compute [tex]\( g(f(x)) \)[/tex]
Next, substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(-2x + 8) \][/tex]
Plug the expression for [tex]\( f(x) \)[/tex] into the function [tex]\( g \)[/tex]:
[tex]\[ g(-2x + 8) = -\frac{1}{2}(-2x + 8) - 8 \][/tex]
Simplify the expression inside the function [tex]\( g \)[/tex]:
[tex]\[ = -\frac{1}{2}(-2x + 8) - 8 \][/tex]
Distribute [tex]\(-\frac{1}{2} \)[/tex] through the parentheses:
[tex]\[ = x - 4 - 8 \][/tex]
Combine like terms:
[tex]\[ = x - 12 \][/tex]
So,
[tex]\[ g(f(x)) = x - 12 \][/tex]
### Conclusion:
Based on the computations, we have:
[tex]\[ f(g(x)) = x + 24 \][/tex]
[tex]\[ g(f(x)) = x - 12 \][/tex]
Since neither [tex]\( f(g(x)) \)[/tex] nor [tex]\( g(f(x)) \)[/tex] simplify to [tex]\( x \)[/tex], the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are not inverses of each other.
Therefore,
No, the given functions are not inverses.
### Step 1: Compute [tex]\( f(g(x)) \)[/tex]
First, substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f\left(-\frac{1}{2}x - 8\right) \][/tex]
Plug the expression for [tex]\( g(x) \)[/tex] into the function [tex]\( f \)[/tex]:
[tex]\[ f\left(-\frac{1}{2}x - 8\right) = -2\left(-\frac{1}{2}x - 8\right) + 8 \][/tex]
Next, simplify the expression inside the function [tex]\( f \)[/tex]:
[tex]\[ = -2 \left(-\frac{1}{2}x - 8\right) + 8 \][/tex]
Distribute [tex]\(-2\)[/tex] through the parentheses:
[tex]\[ = -2 \left(-\frac{1}{2}x \right) - 2(-8) + 8 \][/tex]
[tex]\[ = x + 16 + 8 \][/tex]
Combine like terms:
[tex]\[ = x + 24 \][/tex]
So,
[tex]\[ f(g(x)) = x + 24 \][/tex]
### Step 2: Compute [tex]\( g(f(x)) \)[/tex]
Next, substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(-2x + 8) \][/tex]
Plug the expression for [tex]\( f(x) \)[/tex] into the function [tex]\( g \)[/tex]:
[tex]\[ g(-2x + 8) = -\frac{1}{2}(-2x + 8) - 8 \][/tex]
Simplify the expression inside the function [tex]\( g \)[/tex]:
[tex]\[ = -\frac{1}{2}(-2x + 8) - 8 \][/tex]
Distribute [tex]\(-\frac{1}{2} \)[/tex] through the parentheses:
[tex]\[ = x - 4 - 8 \][/tex]
Combine like terms:
[tex]\[ = x - 12 \][/tex]
So,
[tex]\[ g(f(x)) = x - 12 \][/tex]
### Conclusion:
Based on the computations, we have:
[tex]\[ f(g(x)) = x + 24 \][/tex]
[tex]\[ g(f(x)) = x - 12 \][/tex]
Since neither [tex]\( f(g(x)) \)[/tex] nor [tex]\( g(f(x)) \)[/tex] simplify to [tex]\( x \)[/tex], the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are not inverses of each other.
Therefore,
No, the given functions are not inverses.
