Skills Review for Inverse Functions

Learning Outcomes

  • Solve rational equations by clearing denominators
  • Solve for a variable in a square root equation
  • Convert between radical and exponent notations
  • Solve for a variable in a complex radical or rational exponent equation

The Inverse Functions section covers in great detail everything you need to know about inverse functions. Depending on the type of function you are working with, finding the inverse of a function algebraically can require you to manipulate and rearrange a function’s equation quite a bit. Here we will review some of the techniques that can be used to rearrange certain types of equations and solve for a specified variable.

Manipulate Rational Equations

Equations that contain rational expressions are called rational equations. For example, [latex]\frac{2y+1}{4}=\frac{x}{3}[/latex] is a rational equation (of two variables).

One of the most straightforward ways to solve a rational equation for the indicated variable is to eliminate denominators with the common denominator and then use properties of equality to isolate the indicated variable.

Solve for [latex]y[/latex] in the equation [latex]\frac{1}{2}y-3=2-\frac{3}{4}x[/latex] by clearing the fractions in the equation first.

Multiply both sides of the equation by [latex]4[/latex], the common denominator of the fractional coefficients.

[latex]\begin{array}{c}\frac{1}{2}y-3=2-\frac{3}{4}x\\ 4\left(\frac{1}{2}y-3\right)=4\left(2-\frac{3}{4}x\right)\\\text{}\,\,\,\,4\left(\frac{1}{2}y\right)-4\left(3\right)=4\left(2\right)+4\left(-\frac{3}{4}x\right)\\2y-12=8-3x\\\underline{+12}\,\,\,\,\,\,\underline{+12}\\ 2y=20-3x\\ \frac{2y}{2}=\frac{20-3x}{2} \\ y=10-\frac{3}{2}x\end{array}[/latex]

Example

Solve the equation [latex]\frac{x+5}{8}=\frac{7}{y}[/latex] for [latex]y[/latex].

In the next example, we show how to solve a rational equation with a binomial in the denominator of one term. We will use the common denominator to eliminate the denominators from both fractions. Note that the LCD is the product of both denominators because they do not share any common factors.

Example

Solve the equation [latex]x=\frac{4}{3y+1}[/latex] for [latex]y[/latex].

Manipulate Radical Equations

Radical equations are equations that contain variables in the radicand (the expression under a radical symbol), such as

[latex]\begin{array}{ccc} \sqrt{3y+18}=x & \\ \sqrt{x+3}=y-3 & \\ \sqrt{x+5}-\sqrt{y - 3}=2\end{array}[/latex]

Radical equations are manipulated by eliminating each radical, one at a time until you have solved for the indicated variable.

A General Note: Radical Equations

An equation containing terms with a variable in the radicand is called a radical equation.

How To: Given a radical equation, solve it

  1. Isolate the radical expression containing your variable of interest on one side of the equal sign. Put all remaining terms on the other side.
  2. If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of the equation to the third power. In other words, for an nth root radical, raise both sides to the nth power. Doing so eliminates the radical symbol.
  3. Solve the resulting equation for the variable of interest.
  4. If a radical term still remains, repeat steps 1–2.

Example: Solving an Equation with One Radical

Solve [latex]\sqrt{15 - 2y}=x[/latex] for [latex]y[/latex].

 

Try It

Solve the radical equation: [latex]\sqrt{y+3}=3x - 1[/latex]

Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For example, [latex]{16}^{\frac{1}{2}}[/latex] is another way of writing [latex]\sqrt{16}[/latex] and [latex]{8}^{\frac{2}{3}}[/latex] is another way of writing [latex]\left(\sqrt[3]{8}\right)^2[/latex].

We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals 1. For example, [latex]\frac{2}{3}\left(\frac{3}{2}\right)=1[/latex].

A General Note: Rational Exponents

A rational exponent indicates a power in the numerator and a root in the denominator. There are multiple ways of writing an expression, a variable, or a number with a rational exponent:

[latex]{a}^{\frac{m}{n}}={\left({a}^{\frac{1}{n}}\right)}^{m}={\left({a}^{m}\right)}^{\frac{1}{n}}=\sqrt[n]{{a}^{m}}={\left(\sqrt[n]{a}\right)}^{m}[/latex]

Example: Solving an Equation involving a Variable raised to a Rational Exponent

Solve the equation in which a variable is raised to a rational exponent: [latex]{x}^{\frac{5}{4}}=32[/latex].

Try It

Solve the equation [latex]{x}^{\frac{3}{2}}=125[/latex].

Note: In the two examples above, there was only one variable in the equations. So, we solved for the variable and found a numeric solution. In the case where there are two different variables in the equation and you are asked to solve for one in terms of the other, you still follow the same process as you solve for the indicated variable.