In this section we will utilize function notation and rules of exponents to add, subtract, multiply and divide expressions involving rational exponents. Before we do this, it is important to understand the notation that will be used.
Addition of functions: \(\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)\)
Subtraction of functions: \(\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)\)
Multiplication of functions: \(\left(f\cdot g\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)\)
Division of functions: \(\left(\large\frac{f}{g}\right)\normalsize{\left(x\right)}=\large\frac{f\left(x\right)}{g\left(x\right)}\)
Subtraction of functions: \(\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)\)
Multiplication of functions: \(\left(f\cdot g\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)\)
Division of functions: \(\left(\large\frac{f}{g}\right)\normalsize{\left(x\right)}=\large\frac{f\left(x\right)}{g\left(x\right)}\)
To add or subtract functions, you will combine like terms. Remember, like terms have the same base raised to the same power. If you have like terms, you will add or subtract the coefficients, but leave the exponents unchanged. This is similar to what you did in previous units, except the exponents will now be fractions!
Example 1: Let \(f\left(x\right)=3x\strut^{\frac{2}{3}}\) and \(g\left(x\right)=-5x\strut^{\frac{2}{3}}\)
a) Find \(\left(f+g\right)\left(x\right)\).
\(\begin{align}&\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)\\
&\left(f+g\right)\left(x\right)=3x\strut^{\frac{2}{3}}+-5x\strut^{\frac{2}{3}}\ & \ &\text{1) Substitute in the expressions.}\\
&\left(f+g\right)\left(x\right)=(3+-5)x\strut^{\frac{2}{3}}\ & \ &\text{2) Factor out the greatest common factor.}\\
&\left(f+g\right)\left(x\right)=-2x\strut^{\frac{2}{3}}\ & \ &\text{3) Add the coefficients.}\end{align}\)
&\left(f+g\right)\left(x\right)=3x\strut^{\frac{2}{3}}+-5x\strut^{\frac{2}{3}}\ & \ &\text{1) Substitute in the expressions.}\\
&\left(f+g\right)\left(x\right)=(3+-5)x\strut^{\frac{2}{3}}\ & \ &\text{2) Factor out the greatest common factor.}\\
&\left(f+g\right)\left(x\right)=-2x\strut^{\frac{2}{3}}\ & \ &\text{3) Add the coefficients.}\end{align}\)
b) Find \(\left(f-g\right)\left(x\right)\).
To multiply or divide functions, you will apply rules of exponents. Remember, when you multiply like bases you will add exponents and when you divide like bases, you will subtract exponents. To add or subtract fractions, you will need to find a common denominator and you will add or subtract the numerators.
Example 2: Let \(f\left(x\right)=36x\strut^{\frac{3}{5}}\) and \(g\left(x\right)=4x\strut^{\frac{2}{3}}\)
a) Find \(\left(f\cdot g\right)\left(x\right)\).
\(\begin{align}&\left(f\cdot g\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)\\
&\left(f\cdot g\right)\left(x\right)=36x\strut^{\frac{3}{5}}\cdot4x\strut^{\frac{2}{3}}\ & \ &\text{1) Substitute in the expressions for the functions.}\\
&\left(f\cdot g\right)\left(x\right)=36\cdot4x\strut^{\frac{3}{5}}\cdot x\strut^{\frac{2}{3}}\ & \ &\text{2) Use the commutative property to reorder the terms.}\\
&\left(f\cdot g\right)\left(x\right)=144x\strut^{\frac{3}{5}+\frac{2}{3}}\ & \ &\text{3) Multiply the numbers and add the exponents.}\\
&\left(f\cdot g\right)\left(x\right)=144x\strut^{\frac{9}{15}+\frac{10}{15}}\ & \ &\text{4) Find common denominator for the exponents.}\\
&\left(f\cdot g\right)\left(x\right)=144x\strut^{\frac{19}{15}}\ & \ &\text{5) Add the numerators of the exponents.}\end{align}\)
&\left(f\cdot g\right)\left(x\right)=36x\strut^{\frac{3}{5}}\cdot4x\strut^{\frac{2}{3}}\ & \ &\text{1) Substitute in the expressions for the functions.}\\
&\left(f\cdot g\right)\left(x\right)=36\cdot4x\strut^{\frac{3}{5}}\cdot x\strut^{\frac{2}{3}}\ & \ &\text{2) Use the commutative property to reorder the terms.}\\
&\left(f\cdot g\right)\left(x\right)=144x\strut^{\frac{3}{5}+\frac{2}{3}}\ & \ &\text{3) Multiply the numbers and add the exponents.}\\
&\left(f\cdot g\right)\left(x\right)=144x\strut^{\frac{9}{15}+\frac{10}{15}}\ & \ &\text{4) Find common denominator for the exponents.}\\
&\left(f\cdot g\right)\left(x\right)=144x\strut^{\frac{19}{15}}\ & \ &\text{5) Add the numerators of the exponents.}\end{align}\)
b) Find \(\large\frac{f\left(x\right)}{g\left(x\right)}\).
One other operation with functions we will look at is the composition of functions. A composite function just means that we are combining two rules into one using substitution--you will NOT multiply to find a composite function.
The symbol used for composite functions is \(f\left(g\left(x\right)\right)\) and is read “f of g of x”. An alternative notation is \(f\circ g\left(x\right)\). To evaluate a composite function, the expression for the inner function, \(g\left(x\right)\), is substituted into the outer expression, \(f\left(x\right)\), and then simplified.
Composite Functions will be used later in the unit to prove that two functions are inverses of each other. For now, we will just practice finding composite functions.
Before we do examples with rational exponents, let’s look at composites with linear or quadratic functions.
Example 3: Let \(f\left(x\right)=2x-7\) and \(g\left(x\right)=x^2-x+1\)
a) Find \(f\circ g\left(x\right)\).
\(\begin{align}&f\circ g\left(x\right)=2\left(x^2-x+1\right)-7\ & \ &\text{1) Substitute the expression for}\ g\left(x\right)\ \text{into x in} f\left(x\right).\\
&f\circ g\left(x\right)=\left(2x^2-2x+2\right)-7\ & \ &\text{2) Distribute. }\\
&f\circ g\left(x\right)=2x^2-2x-5\ & \ &\text{3) Combine like terms.}\end{align}\)
\(\begin{align}&f\circ g\left(x\right)=2\left(x^2-x+1\right)-7\ & \ &\text{1) Substitute the expression for}\ g\left(x\right)\ \text{into x in} f\left(x\right).\\
&f\circ g\left(x\right)=\left(2x^2-2x+2\right)-7\ & \ &\text{2) Distribute. }\\
&f\circ g\left(x\right)=2x^2-2x-5\ & \ &\text{3) Combine like terms.}\end{align}\)
b) Find \(g\left(f\left(x\right)\right)\). c) Find \(f\left(g\left(3\right)\right)\) and \(g\left(f\left(3\right)\right)\).
The video below explains examples b and c.
Composition of functions is NOT commutative--that is \(f\left(g\left(x\right)\right)\ne g\left(f\left(x\right)\right)\).
Example 4: Let \(f\left(x\right)=4x\strut^{\frac{2}{3}}\) and \(g\left(x\right)=8x\strut^{\frac{4}{5}}\). Find \(f\left(g\left(x\right)\right)\).
\(\begin{align}&f\left(g\left(x\right)\right)=4\left(8x\strut^{\frac{4}{5}}\right)^{\frac{2}{3}}\ & \ &\text{1) Substitute the expression for} g\left(x\right)\ \text{into}\ x\ \text{in}\ f\left(x\right).\\
&f\left(g\left(x\right)\right)=4\left(8\strut^{\frac{2}{3}}\cdot x\strut^{\frac{4}{5}\cdot\frac{2}{3}}\right)\ & \ &\text{2) Distribute the exponent to each term.}\\
&f\left(g\left(x\right)\right)=4\left(4\cdot x\strut^{\frac{8}{15}}\right)\ & \ &\text{3) Perform the operations with rational exponents.}\\
&f\left(g\left(x\right)\right)=16x\strut^{\frac{8}{15}}\ & \ &\text{4) Multiply.}\end{align}\)
&f\left(g\left(x\right)\right)=4\left(8\strut^{\frac{2}{3}}\cdot x\strut^{\frac{4}{5}\cdot\frac{2}{3}}\right)\ & \ &\text{2) Distribute the exponent to each term.}\\
&f\left(g\left(x\right)\right)=4\left(4\cdot x\strut^{\frac{8}{15}}\right)\ & \ &\text{3) Perform the operations with rational exponents.}\\
&f\left(g\left(x\right)\right)=16x\strut^{\frac{8}{15}}\ & \ &\text{4) Multiply.}\end{align}\)
Example 5: Let \(f\left(x\right)=25x\strut^{\frac{2}{3}}\) and \(g\left(x\right)=x\strut^{\frac{5}{2}}\). Find \(g\left(f\left(x\right)\right)\).