Sure! Let's find the inverse of each function step by step.
### Finding the inverse of [tex]\( f(x) = -2x + 11 \)[/tex]
1. Start with the function:
[tex]\[
f(x) = -2x + 11
\][/tex]
2. To find the inverse, we first replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = -2x + 11
\][/tex]
3. Next, interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = -2y + 11
\][/tex]
4. Solve this equation for [tex]\( y \)[/tex]:
[tex]\[
x = -2y + 11
\][/tex]
[tex]\[
x - 11 = -2y
\][/tex]
[tex]\[
-2y = x - 11
\][/tex]
[tex]\[
y = \frac{11 - x}{2}
\][/tex]
5. Replace [tex]\( y \)[/tex] with [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = \frac{11 - x}{2}
\][/tex]
So, the inverse function of [tex]\( f(x) = -2x + 11 \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{11 - x}{2}
\][/tex]
### Finding the inverse of [tex]\( g(x) = \frac{4}{x + 7} \)[/tex]
1. Start with the function:
[tex]\[
g(x) = \frac{4}{x + 7}
\][/tex]
2. To find the inverse, replace [tex]\( g(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = \frac{4}{x + 7}
\][/tex]
3. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \frac{4}{y + 7}
\][/tex]
4. Solve this equation for [tex]\( y \)[/tex]:
[tex]\[
x(y + 7) = 4
\][/tex]
[tex]\[
xy + 7x = 4
\][/tex]
[tex]\[
xy = 4 - 7x
\][/tex]
[tex]\[
y = \frac{4 - 7x}{x}
\][/tex]
5. Replace [tex]\( y \)[/tex] with [tex]\( g^{-1}(x) \)[/tex]:
[tex]\[
g^{-1}(x) = \frac{4 - 7x}{x}
\][/tex]
So, the inverse function of [tex]\( g(x) = \frac{4}{x + 7} \)[/tex] is:
[tex]\[
g^{-1}(x) = \frac{4 - 7x}{x}
\][/tex]
In summary:
1. The inverse of [tex]\( f(x) = -2x + 11 \)[/tex] is [tex]\( f^{-1}(x) = \frac{11 - x}{2} \)[/tex].
2. The inverse of [tex]\( g(x) = \frac{4}{x + 7} \)[/tex] is [tex]\( g^{-1}(x) = \frac{4 - 7x}{x} \)[/tex].
