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High School Math : Understanding Inverse Functions

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Example Questions

Example Question #1 : Understanding Inverse Functions

Let $\displaystyle f(x)=5-(1+x)^3$ . What is $f^{-1}(x)$ $\displaystyle f^{-1}(x)$ ?

Possible Answers:

$f^{-1}(x)=(5-x)^{\frac{1}{3}}-1$ $\displaystyle f^{-1}(x)=(5-x)^{\frac{1}{3}}-1$

$f^{-1}(x)=\left (1-x)\right ^{1/3}+5$

$f^{-1}(x)=(5-x)^{3}-1$ $\displaystyle f^{-1}(x)=(5-x)^{3}-1$

$f^{-1}(x)=\left ((5-x)^{3}-1 )\right ^{1/3}$

$f^{-1}(x)=\left (1-5x)\right ^{1/3}+1$

Correct answer:

$f^{-1}(x)=(5-x)^{\frac{1}{3}}-1$ $\displaystyle f^{-1}(x)=(5-x)^{\frac{1}{3}}-1$

Explanation:

We are asked to find $f^{^{-1}} (x)$ $\displaystyle f^{^{-1}} (x)$ , which is the inverse of a function.

In order to find the inverse, the first thing we want to do is replace f(x) with y. (This usually makes it easier to separate x from its function.).

$\displaystyle y=5-(1+x)^3$

Next, we will swap x and y.

$\displaystyle x=5-(1+y)^3$

Then, we will solve for y. The expression that we determine will be equal to $f^{^{-1}} (x)$ $\displaystyle f^{^{-1}} (x)$ .

$\displaystyle x=5-(1+y)^3$

Subtract 5 from both sides.

$\displaystyle x-5=-(1+y)^3$

Multiply both sides by -1.

$\displaystyle -1(x-5)=5-x=(1+y)^3$

We need to raise both sides of the equation to the 1/3 power in order to remove the exponent on the right side.

$(5-x)^{\frac{1}{3}}=\left ((1+y)^{3} \right )^{1/3}$ $\displaystyle (5-x)^{\frac{1}{3}}=\left ((1+y)^{3} \right )^{1/3}$

We will apply the general property of exponents which states that $(a^b)^{c}=a^{b\cdot c}$ $\displaystyle (a^b)^{c}=a^{b\cdot c}$ .

$(5-x)^{\frac{1}{3}}=\left ((1+y)^{3} \right )^{1/3}=(1+y)^{3\cdot \frac{1}{3}}=(1+y)^1=1+y$ $\displaystyle (5-x)^{\frac{1}{3}}=\left ((1+y)^{3} \right )^{1/3}=(1+y)^{3\cdot \frac{1}{3}}=(1+y)^1=1+y$

Laslty, we will subtract one from both sides.

$(5-x)^{\frac{1}{3}}-1=y$ $\displaystyle (5-x)^{\frac{1}{3}}-1=y$

The expression equal to y is equal to the inverse of the original function f(x). Thus, we can replace y with $f^{-1}(x)$ $\displaystyle f^{-1}(x)$ .

$f^{-1}(x)=(5-x)^{\frac{1}{3}}-1$ $\displaystyle f^{-1}(x)=(5-x)^{\frac{1}{3}}-1$

The answer is $f^{-1}(x)=(5-x)^{\frac{1}{3}}-1$ $\displaystyle f^{-1}(x)=(5-x)^{\frac{1}{3}}-1$ .

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Example Question #1 : Understanding Inverse Functions

What is the inverse of $y=4x^{2}$ $\displaystyle y=4x^{2}$ ?

Possible Answers:

$y=\frac{\pm \sqrt{x}}{4}$ $\displaystyle y=\frac{\pm \sqrt{x}}{4}$

$y=-4x^{2}$ $\displaystyle y=-4x^{2}$

$y=\frac{\pm \sqrt{x}}{2}$ $\displaystyle y=\frac{\pm \sqrt{x}}{2}$

$y=\frac{x^{2}}{4}$ $\displaystyle y=\frac{x^{2}}{4}$

$y=\pm \sqrt{x}$ $\displaystyle y=\pm \sqrt{x}$

Correct answer:

$y=\frac{\pm \sqrt{x}}{2}$ $\displaystyle y=\frac{\pm \sqrt{x}}{2}$

Explanation:

The inverse of $\displaystyle y$ requires us to interchange $\displaystyle x$ and $\displaystyle y$ and then solve for $\displaystyle y$ .

$y=4x^{2}$ $\displaystyle y=4x^{2}$

$x=4y^{2}$ $\displaystyle x=4y^{2}$

Then solve for $\displaystyle y$ :

$y=\frac{\pm \sqrt{x}}{2}$ $\displaystyle y=\frac{\pm \sqrt{x}}{2}$

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Example Question #371 : Algebra Ii

If $f(x)=x^{2}+1$ $\displaystyle f(x)=x^{2}+1$ , what is $f^{-1}(x)$ $\displaystyle f^{-1}(x)$ ?

Possible Answers:

$f^{-1}(x)=\frac{1}{x^{2}-1}$ $\displaystyle f^{-1}(x)=\frac{1}{x^{2}-1}$

$f^{-1}(x)=x^{2}-1$ $\displaystyle f^{-1}(x)=x^{2}-1$

$f^{-1}(x)=\sqrt{x-1}$ $\displaystyle f^{-1}(x)=\sqrt{x-1}$

$f^{-1}(x)=\sqrt{x^{2}+1}$ $\displaystyle f^{-1}(x)=\sqrt{x^{2}+1}$

$f^{-1}(x)=\frac{1}{x^{2}+1}$ $\displaystyle f^{-1}(x)=\frac{1}{x^{2}+1}$

Correct answer:

$f^{-1}(x)=\sqrt{x-1}$ $\displaystyle f^{-1}(x)=\sqrt{x-1}$

Explanation:

To find the inverse of a function, exchange the $\displaystyle x$ and $\displaystyle y$ variables and then solve for $\displaystyle y$ .

$x=y^{2}+1$ $\displaystyle x=y^{2}+1$

$x-1=y^{2}$ $\displaystyle x-1=y^{2}$

$\sqrt{x-1}=y=f^{-1}(x)$ $\displaystyle \sqrt{x-1}=y=f^{-1}(x)$

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High School Math : Understanding Inverse Functions

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Understanding Inverse Functions

Example Question #1 : Understanding Inverse Functions

Example Question #371 : Algebra Ii

All High School Math Resources

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