Inverse Functions - Pre-Calculus

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Question

Find the inverse of the following equation:

Answer

To find the inverse of a function, replace the x any y positions:

Original Equation:

Inversed Equation: $x = 2(y_{inv}^3+12)+3$

Now solve for the inversed y value.

$x = 2(y_{inv}^3+12)+3$

$x-3 = 2(y_{inv}^3+12)$

$\frac{x-3}{2}= y_{inv}^3+12$

$\frac{x-3}{2}-12= y_{inv}^3$

$y_{inv}=(\frac{x-3}{2}-12)^\frac{1}{3} = (\frac{x-27}{2})^\frac{1}{3}$

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Question

What is the inverse function of

?

Answer

To find the inverse function of

we replace the with and vice versa.

So

Now solve for

$\frac{3y}{3}=\frac{x-5}{3}$

$y=\frac{x-5}{3}$

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Question

What is the inverse of $\left(\frac{1}{2},3\right)$ ?

Answer

When trying to find the inverse of a point, switch the x and y values.

So, $\left(\frac{1}{2},3\right)\rightarrow \left(3, \frac{1}{2} \right )$

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Question

Find the inverse of,

.

Answer

In order to find the inverse, switch the x and y variables in the function then solve for y.

Switching variables we get,

.

Then solving for y to get our final answer.

$y=\frac{x-1}{3}$

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Question

Find the inverse of .

Answer

To find the inverse of this function we can switch the and variables and solve for .

First, switch and in the function:

Now, solve for :

$y = \pm \sqrt{x+2}$

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Question

Find the inverse of,

$y=x^{2}$ .

Answer

First, switch the variables making into $x=y^{2}$ .

Then solve for y by taking the square root of both sides.

$\sqrt {y^2}=\pm \sqrt x$

$y=\pm \sqrt x$

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Question

Find the inverse of the following equation.

$y=\sqrt{\ln(x)+2}$ .

Answer

To find the inverse in this case, we need to switch our x and y variables and then solve for y.

Therefore,

$y=\sqrt{\ln(x)+2}$ becomes,

$x=\sqrt{\ln(y)+2}$

To solve for y we square both sides to get rid of the sqaure root.

$x^2=\ln(y)+2$

We then subtract 2 from both sides and take the exponenetial of each side, leaving us with the final answer.

$x^2-2=\ln(y)$

$e^{x^2-2}=e^{\ln(y)}$

$e^x^{^{2}-2}=y$

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Question

Find the inverse of the following function.

Answer

To find the inverse of y, or

$y^{-1}$

first switch your variables x and y in the equation.

$x_{i}=y_i^3+e^0$

Second, solve for the variable in the resulting equation.

$(x_i-e^0)^\frac{1}{3}=(y_i^{3})^\frac{1}{3}$

$y_i=\sqrt[3]{x_i-e^0}$

Simplifying a number with 0 as the power, the inverse is

$y^{-1}=\sqrt[{3}]{x-1}$

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Question

Find the inverse of the following function.

$y=\ln, x: +: \pi$

Answer

To find the inverse of y, or

$y^{-1}$

first switch your variables x and y in the equation.

$x_i=\ln(y_i)+\pi$

Second, solve for the variable in the resulting equation.

$x_i-\pi=\ln,y_i$

And by setting each side of the equation as powers of base e,

$y^{-1}=e^{x-\pi}$

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Question

Find the inverse of the function.

Answer

To find the inverse we need to switch the variables and then solve for y.

Switching the variables we get the following equation,

.

Now solve for y.

$y^{-1}=x-5$

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Question

If $f(x)=-\frac{3x}{5}+11$ , what is its inverse function, $f^{-1}(x)$ ?

Answer

We begin by taking $f(x)=-\frac{3x}{5}+11$ and changing the to a , giving us $y=-\frac{3x}{5}+11$ .

Next, we switch all of our and , giving us $x=-\frac{3y}{5}+11$ .

Finally, we solve for by subtracting from each side, multiplying each side by , and dividing each side by , leaving us with,

$y=-\frac{5x}{3}+\frac{55}{3}$ .

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Question

Find the inverse of

Answer

So we first replace every with an and every with a .

Our resulting equation is:

Now we simply solve for y.

Subtract 9 from both sides:

Now divide both sides by 10:

$\frac{x-9}{10}=\frac{10y}{10}$

$\frac{x-9}{10}=y$

The inverse of

is

$y=\frac{x-9}{10}$

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Question

What is the inverse of

Answer

To find the inverse of a function we just switch the places of all and with eachother.

So

turns into

Now we solve for

Divide both sides by

$\frac{x}{9}=\frac{9y}{9}$

$y=\frac{x}{9}$

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Question

Find the inverse of the follow function:

Answer

To find the inverse, substitute all x's for y's and all y's for x's and then solve for y.

$\frac{x+9}{3}=y^2$

$y=\pm\sqrt{\frac{x+9}{3}}$

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Question

Find the inverse of .

Answer

To find the inverse of the function, we switch the switch the and variables in the function.

Switching and gives

Then, solving for gives our answer:

$\frac{x-1}{10} = y$

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Question

Find the inverse of .

Answer

To find the inverse of the function, we must swtich and variables in the function.

Switching and gives:

Solving for yields our final answer:

$y= \frac{x-2}{2}$

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Question

Find the inverse of .

Answer

To find the inverse of the function, we can switch and in the function and solve for :

Switching and gives:

Solving for yields our final answer:

$y = \frac{x+3}{4}$

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Question

Find the inverse of .

Answer

To find the inverse of the function, we can switch and in the function and solve for .

Switch and :

We can now solve for :

$y = \frac{x+3}{7}$

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Question

Find the inverse of .

Answer

To find the inverse of the function, we simply need to switch the values of and and solve for .

Switching and , we can write the function as:

We now subtract to solve for :

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Question

Find the inverse of .

Answer

To find the inverse of this function, we switch and in the function:

We now solve for :

$y = \frac{x+7}{3}$

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