To find the inverse of the function [tex]\( f(x) = \sqrt{1 - x} + 7 \)[/tex] and determine the domain of the inverse, we can follow these steps:
1. Write the function equation: [tex]\( y = \sqrt{1 - x} + 7 \)[/tex].
2. Isolate the square root term: Subtract 7 from both sides to isolate the square root term.
[tex]\[
y - 7 = \sqrt{1 - x}
\][/tex]
3. Square both sides: To eliminate the square root, square both sides of the equation.
[tex]\[
(y - 7)^2 = 1 - x
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 1 - (y - 7)^2
\][/tex]
5. Express the inverse function: The inverse function [tex]\( f^{-1}(x) \)[/tex] is given by:
[tex]\[
f^{-1}(x) = 1 - (x - 7)^2
\][/tex]
Next, we need to determine the domain of the inverse function [tex]\( f^{-1}(x) \)[/tex].
The domain of the inverse function will be the range of the original function [tex]\( f(x) = \sqrt{1 - x} + 7 \)[/tex].
- For [tex]\( f(x) \)[/tex], the domain is [tex]\( (-\infty, 1] \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( \sqrt{1 - x} \)[/tex] becomes large, but the function is only defined up to [tex]\( x = 1 \)[/tex].
- At [tex]\( x = 1 \)[/tex], the value of [tex]\( f(x) \)[/tex] is:
[tex]\[
f(1) = \sqrt{1 - 1} + 7 = 0 + 7 = 7
\][/tex]
- As [tex]\( x \)[/tex] moves towards [tex]\( -\infty \)[/tex], [tex]\( \sqrt{1 - x} \)[/tex] increases indefinitely, and hence, [tex]\( f(x) \)[/tex] can take any value greater than or equal to 7.
Therefore, the range of [tex]\( f(x) \)[/tex], which becomes the domain of [tex]\( f^{-1}(x) \)[/tex], is:
[tex]\[
[7, \infty)
\][/tex]
In conclusion:
- The inverse function is [tex]\( f^{-1}(x) = 1 - (x - 7)^2 \)[/tex].
- The domain of [tex]\( f^{-1}(x) \)[/tex] is [tex]\([7, \infty)\)[/tex].
