To determine whether the given pairs of functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other, we need to compute [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex]. If both compositions simplify to [tex]\( x \)[/tex], then [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are indeed inverses of each other.
### Pair 1
Given:
- [tex]\( f(x) = -\frac{x}{6} \)[/tex]
- [tex]\( g(x) = -6x \)[/tex]
Finding [tex]\( f(g(x)) \)[/tex]:
[tex]\[
f(g(x)) = f(-6x) = -\frac{-6x}{6} = x
\][/tex]
Finding [tex]\( g(f(x)) \)[/tex]:
[tex]\[
g(f(x)) = g\left(-\frac{x}{6}\right) = -6\left(-\frac{x}{6}\right) = x
\][/tex]
Since both [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] simplify to [tex]\( x \)[/tex], the functions [tex]\( f(x) = -\frac{x}{6} \)[/tex] and [tex]\( g(x) = -6x \)[/tex] are inverses of each other.
### Pair 2
Given:
- [tex]\( f(x) = 2x + 7 \)[/tex]
- [tex]\( g(x) = \frac{x - 7}{2} \)[/tex]
Finding [tex]\( f(g(x)) \)[/tex]:
[tex]\[
f(g(x)) = f\left(\frac{x - 7}{2}\right) = 2\left(\frac{x - 7}{2}\right) + 7 = (x - 7) + 7 = x
\][/tex]
Finding [tex]\( g(f(x)) \)[/tex]:
[tex]\[
g(f(x)) = g(2x + 7) = \frac{(2x + 7) - 7}{2} = \frac{2x}{2} = x
\][/tex]
Since both [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] simplify to [tex]\( x \)[/tex], the functions [tex]\( f(x) = 2x + 7 \)[/tex] and [tex]\( g(x) = \frac{x - 7}{2} \)[/tex] are inverses of each other.
### Conclusion
For both pairs of functions, the compositions simplify to [tex]\( x \)[/tex]. Thus:
- For [tex]\( f(x) = -\frac{x}{6} \)[/tex] and [tex]\( g(x) = -6x \)[/tex], [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other.
- For [tex]\( f(x) = 2x + 7 \)[/tex] and [tex]\( g(x) = \frac{x - 7}{2} \)[/tex], [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other.
