Question

show steps to solve. 8) find the inverse of the function ( f(x) = sqrt{x} - 13 ) algebraically.
change ( f(x) ) to ( y )
switch ( x ) and ( y ).
solve for ( y )
answer -> change ( y ) to ( f^{-1}(x) )

Explanation:

Step1: Change \( f(x) \) to \( y \)

We start with the function \( f(x)=\sqrt{x}-13 \). Replace \( f(x) \) with \( y \), so we get \( y = \sqrt{x}-13 \).

Step2: Switch \( x \) and \( y \)

Interchange the roles of \( x \) and \( y \) in the equation. So the equation becomes \( x=\sqrt{y}-13 \).

Step3: Solve for \( y \)

First, add 13 to both sides of the equation to isolate the square root term. We have \( x + 13=\sqrt{y} \). Then, to eliminate the square root, we square both sides of the equation. Squaring the left - hand side gives \( (x + 13)^2 \) and squaring the right - hand side gives \( y \). So \( y=(x + 13)^2 \). We can expand \( (x + 13)^2 \) using the formula \( (a + b)^2=a^{2}+2ab + b^{2} \), where \( a=x \) and \( b = 13 \). So \( (x + 13)^2=x^{2}+26x + 169 \), and \( y=x^{2}+26x + 169 \). We should also note the domain of the inverse function. The original function \( f(x)=\sqrt{x}-13 \) has a domain \( x\geq0 \) and a range \( y\geq - 13 \). For the inverse function, the domain is \( x\geq - 13 \) and the range is \( y\geq0 \).

Step4: Change \( y \) to \( f^{-1}(x) \)

Replace \( y \) with \( f^{-1}(x) \) in the equation \( y=(x + 13)^2 \) (with the domain \( x\geq - 13 \)).

Answer:

The inverse function is \( f^{-1}(x)=(x + 13)^2 \), where \( x\geq - 13 \)