Question

2 parts

What is the inverse of f(x) = 6x -24 and second question is


Find the inverse of g(x) = 3x^2 - 5

Answer 1

Answer with step-by-step explanation:

1) Inverse of $f(x) = 6x -24$:

Make the function equal to y to get $y=6x-24$.

Now making $x$ the subject of the function:

$6x=y+24\\\\x=\frac{y+24}{6} \\\\x=\frac{y}{6} +4$

Change back the variable $y$ to $x$ and this is the inverse.

$f'(x)=\frac{x}{6} +4$

2. Inverse of $g(x) = 3x^2 - 5$:

Making the function equal to y to get: $y=3x^2 - 5$

Now making $x$ the subject of the function:

$3x^2=y+5\\\\x^2=\frac{y+5}{3}$

Taking square root on both sides to get:

$x=\sqrt{\frac{y+5}{3} }$

Change back the variable $y$ to $x$ and this is the inverse.

$g'(x)=\sqrt{\frac{x+5}{3} }$

Answer 2

Answer:

1) $f^{-1}(x)=\frac{x}{6}+4$

2)  $g^{-1}(x)=\sqrt{\frac{x+5}{3}}$

Step-by-step explanation:

1) To find the inverse of the function $f(x) = 6x -24$ you need to follow these steps:

- Since $f(x)=y$, you can rewrite the function:

 $y = 6x -24$

- Solve for "x":

$y+24=6x\\\\x=\frac{y+24}{6}\\\\x=\frac{y}{6}+4$

- Exchange the variables:

$y=\frac{x}{6}+4$

Then, the inverse is:

$f^{-1}(x)=\frac{x}{6}+4$

2) To find the inverse of the function $g(x) = 3x^2 - 5$ you need to follow these steps:

- Since $g(x)=y$, you can rewrite the function:

 $y= 3x^2 - 5$

- Solve for "x":

$y= 3x^2 - 5\\\\\frac{y+5}{3}=x^2\\\\x=\sqrt{\frac{y+5}{3}}$

- Exchange the variables:

$y=\sqrt{\frac{x+5}{3}}$

 Then, the inverse is:

$g^{-1}(x)=\sqrt{\frac{x+5}{3}}$