Question

SS below:

Answer 1

To sum up the concept of inverse function

If you have function f(x)

if f(\(\color{orange}{x}\)) = \(\color{red}{a}\)

then your inverse function \( f^{-1}(x)\)

would be
\( f^{-1}(\color{red}{a}) = \color{orange}{x}\)

the x-value of the inverse function is going to be the y-value of the normal function

the y-value of the inverse function is going to be the x-value of the normal function

Answer 2

To find the inverse function however, you have to follow a few steps

first change f(x) to x
and change the x to y

you should know that f(x) is the same thing as y
and when we're trying to find the inverse, we are switching the x to a y and the y to an x

Once you do that, you have to solve the equation for y
and then you can just replace y with \( f^{-1}(x)\)

Answer 3

One step at a time

\(\color{red}{f(x)} = 6\color{blue}{x}^2 - 7\)

replace the red f(x) with 'x'

and replace the blue x with 'y'

What do you get?

Answer 4

x=6y^2-7

Answer 5

Good

Now, can you solve for y?

Answer 6

\[\pm \sqrt{\frac{ x+7 }{ 6 }}\]

Answer 7

Right, and we'll only take the positive value for it because I'm assuming the original function has a constraint of x≥0

and so now all you have to do is replace y with \( f^{-1}(x)\) and it equals to\(\sqrt{\dfrac{x+7}{6}}\)

Answer 8

thank you

Answer 9

no problem!