To determine which function is the inverse of [tex]\( f(x) = -5x - 4 \)[/tex], let's consider each candidate [tex]\( f^{-1}(x) \)[/tex] and verify whether it satisfies the property of an inverse function: [tex]\( f(f^{-1}(x)) = x \)[/tex].
1. First candidate: [tex]\( f^{-1}(x) = -\frac{1}{5}x - \frac{4}{5} \)[/tex]
We need to check if [tex]\( f(f^{-1}(x)) = x \)[/tex]:
[tex]\[ f\left( -\frac{1}{5}x - \frac{4}{5} \right) = -5 \left( -\frac{1}{5}x - \frac{4}{5} \right) - 4 \][/tex]
[tex]\[ = -5 \left( -\frac{1}{5}x \right) - 5 \left( -\frac{4}{5} \right) - 4 \][/tex]
[tex]\[ = x + 4 - 4 \][/tex]
[tex]\[ = x \][/tex]
Therefore, [tex]\( f(f^{-1}(x)) = x \)[/tex] holds true, making this function a valid candidate for the inverse.
2. Second candidate: [tex]\( f^{-1}(x) = -\frac{1}{5}x + \frac{4}{5} \)[/tex]
We need to check if [tex]\( f(f^{-1}(x)) = x \)[/tex]:
[tex]\[ f\left( -\frac{1}{5}x + \frac{4}{5} \right) = -5 \left( -\frac{1}{5}x + \frac{4}{5} \right) - 4 \][/tex]
[tex]\[ = -5 \left( -\frac{1}{5}x \right) + 5 \left( \frac{4}{5} \right) - 4 \][/tex]
[tex]\[ = x + 4 - 4 \][/tex]
[tex]\[ = x \][/tex]
Therefore, [tex]\( f(f^{-1}(x)) = x \)[/tex] holds true, making this function another valid candidate for the inverse.
3. Third candidate: [tex]\( f^{-1}(x) = -4x + 5 \)[/tex]
We need to check if [tex]\( f(f^{-1}(x)) = x \)[/tex]:
[tex]\[ f\left( -4x + 5 \right) = -5 \left( -4x + 5 \right) - 4 \][/tex]
[tex]\[ = 20x - 25 - 4 \][/tex]
[tex]\[ = 20x - 29 \][/tex]
This expression does not simplify to [tex]\( x \)[/tex]. Therefore, [tex]\( f(f^{-1}(x)) \neq x \)[/tex], making this function not a valid inverse.
4. Fourth candidate: [tex]\( f^{-1}(x) = 4x + 4 \)[/tex]
We need to check if [tex]\( f(f^{-1}(x)) = x \)[/tex]:
[tex]\[ f\left( 4x + 4 \right) = -5 \left( 4x + 4 \right) - 4 \][/tex]
[tex]\[ = -20x - 20 - 4 \][/tex]
[tex]\[ = -20x - 24 \][/tex]
This expression does not simplify to [tex]\( x \)[/tex]. Therefore, [tex]\( f(f^{-1}(x)) \neq x \)[/tex], making this function also not a valid inverse.
Conclusion:
Since none of the functions [tex]\( f^{-1}(x) = -\frac{1}{5}x - \frac{4}{5} \)[/tex], [tex]\( f^{-1}(x) = -\frac{1}{5}x + \frac{4}{5} \)[/tex], [tex]\( f^{-1}(x) = -4x + 5 \)[/tex], and [tex]\( f^{-1}(x) = 4x + 4 \)[/tex] satisfies [tex]\( f(f^{-1}(x)) = x \)[/tex], the provided candidates do not correctly represent the inverse of [tex]\( f(x)=-5x-4 \)[/tex].
Thus, the result is that none of the given functions is the inverse of [tex]\( f(x) = -5x - 4 \)[/tex].
