Question

evaluate independent practice
learning goal
i can determine the inverse of a function by solving f(x)=c. i can determine by composition that one function is the inverse of another, f(g(x))=x. i can determine the values of the inverse function from a graph or a table. i can describe the domain that will produce an invertible function from a non - invertible function.
lesson 10.1 checkpoint
once you have completed the above problems and checked your solutions, complete the lesson checkpoint below.
complete the lesson reflection above by circling your current understanding of the learning goal.
what is the inverse of the given function?

1. f(x)=x² - 6
2. g(x)=-12x³ + 5

Explanation:

Step1: Replace $f(x)$ with $y$ for $f(x)=x^{2}-6$

$y = x^{2}-6$

Step2: Swap $x$ and $y$

$x = y^{2}-6$

Step3: Solve for $y$

$y^{2}=x + 6$
$y=\pm\sqrt{x + 6}$
Since the original function $f(x)=x^{2}-6$ is not one - to - one on the entire real line, if we consider the domain $x\geq0$ for $f(x)$, the inverse is $f^{-1}(x)=\sqrt{x + 6},x\geq - 6$.

Step4: For $g(x)=-12x^{3}+5$, replace $g(x)$ with $y$

$y=-12x^{3}+5$

Step5: Swap $x$ and $y$

$x=-12y^{3}+5$

Step6: Solve for $y$

$12y^{3}=5 - x$
$y^{3}=\frac{5 - x}{12}$
$y=\sqrt[3]{\frac{5 - x}{12}}$

Answer:

1. If considering the domain $x\geq0$ for $f(x)=x^{2}-6$, the inverse is $f^{-1}(x)=\sqrt{x + 6},x\geq - 6$
2. The inverse of $g(x)=-12x^{3}+5$ is $g^{-1}(x)=\sqrt[3]{\frac{5 - x}{12}}$