Lesson 20: Performing Operations with Functions

New Concept

You can perform basic arithmetic operations on functions. For addition, the notation is: $(f + g)(x) = f(x) + g(x)$.

What’s next

Next, you'll apply these operations numerically, algebraically, and graphically to combine functions and solve problems.

Property

Functions can be combined using standard arithmetic operations to create entirely new functions. The notation tells you exactly which operation to perform on the outputs of the original functions:

• Addition: $(f + g)(x) = f(x) + g(x)$
• Subtraction: $(f - g)(x) = f(x) - g(x)$
• Multiplication: $(fg)(x) = f(x) \cdot g(x)$
• Division: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, where $g(x) \neq 0$

To evaluate a combined function at a specific number (like $x = 3$), you can either combine the algebraic expressions first and then plug in the number, or evaluate $f(3)$ and $g(3)$ separately and then combine their numerical results.

Examples

• Algebraic vs. Numerical: Let $f(x) = x + 5$ and $g(x) = 2x$. Find $(f + g)(3)$.

Numerical method: $f(3) = 8$ and $g(3) = 6$. Add the results: $8 + 6 = 14$.
Algebraic method: $(f + g)(x) = (x + 5) + 2x = 3x + 5$. Substitute 3: $3(3) + 5 = 14$.

• Division Domain Constraint: Let $f(x) = x + 4$ and $g(x) = x - 2$. Find $(\frac{f}{g})(3)$.

First, verify the denominator is not zero: $g(3) = 3 - 2 = 1$.
Evaluate the numerator: $f(3) = 3 + 4 = 7$.
The quotient is $\frac{7}{1} = 7$.

To find the sum or difference of two functions, you can work numerically by evaluating each function at a specific value first, or work algebraically by combining the function expressions. For example, $(h+g)(2) = h(2) + g(2)$ is the numerical method, while $(h+g)(x) = h(x) + g(x)$ is the algebraic method.

Given $f(x)=x+5$ and $g(x)=2x$:

• Find $(f+g)(3)$ numerically: $f(3)=8$, $g(3)=6$, so $8+6=14$.
• Find $(f-g)(x)$ algebraically: $(f-g)(x) = (x+5) - 2x = -x+5$.
• Find $(f-g)(10)$ using the algebraic result: $-(10)+5 = -5$.

It's like paying for snacks with a friend. You can either figure out your individual costs first and then add them up (numerical), or you can put all the snacks on one bill and then calculate the total (algebraic). Both methods get you to the same total, so you can choose whichever path seems easier for the problem!

To multiply functions, find the product of their expressions: $(fg)(x) = f(x) \cdot g(x)$. To divide, create a fraction with the functions: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$. An essential rule for division is that the domain cannot include any value of $x$ that makes the denominator, $g(x)$, equal to zero.

Given $h(x)=x+3$ and $g(x)=x-6$:

• Find $(hg)(-4)$ numerically: $h(-4)=-1$, $g(-4)=-10$. So $(-1)(-10)=10$.
• Find $(\frac{h}{g})(x)$ algebraically: $(\frac{h}{g})(x) = \frac{x+3}{x-6}$, where $x \neq 6$.
• Find $(\frac{h}{g})(7)$ numerically: $h(7)=10$, $g(7)=1$. So $(\frac{h}{g})(7) = \frac{10}{1} = 10$.

Multiplying functions is like finding the area of a field where the lengths of the sides are defined by your functions. When you divide functions, it's like splitting treasure, but you must make sure the number of pirates you're dividing by isn't zero! If $g(x)=0$, the division is undefined, and the math treasure map leads nowhere.

In business, a profit function $p(x)$ can be created by subtracting the cost function $f(x)$ from the income (or revenue) function $g(x)$. The formula is:

A baker's cost is $f(x) = 3 + 6.25x$ dollars and income is $g(x)=12x$ dollars.

• The profit function is $p(x) = 12x - (3 + 6.25x) = 5.75x - 3$ dollars.
• The profit from selling 6 pastries is $p(6) = 5.75(6) - 3 = 34.50 - 3 = 31.50$ dollars.

Imagine you're selling custom sneakers. Your income is all the money you collect, but you had to buy the plain shoes and paint first. That's your cost. To find out if you're actually making money, you subtract your costs from your income. This new 'profit function' is your business's report card, telling you exactly how much you earn per sneaker.