For $f(x) = 3x - 2$ and $g(x) = x^2 + 1$, find $(f + g)(x)$ and $(f - g)(2)$.

$(f + g)(x) = x^2 + 3x - 1$; $(f - g)(2) = -1$

$f = \{(-1, 4), (0, 2), (2, 5), (3, 1)\}$ and $g = \{(-2, 3), (0, -1), (2, 4), (3, 6)\}$. Find $f + g$ and $fg$.

$f + g = \{(0, 1), (2, 9), (3, 7)\}$; $fg = \{(0, -2), (2, 20), (3, 6)\}$

For $f(x) = \sqrt{x - 1}$ and $g(x) = \sqrt{5 - x}$, find the domain of $f + g$ and of $fg$, and evaluate $(fg)(3)$.

Both have domain $1 \le x \le 5$; $(fg)(3) = 2$

For $f(x) = x^2 - 9$ and $g(x) = x^2 - x - 6$, simplify $\left(\frac{f}{g}\right)(x)$ and state its domain.

$\left(\frac{f}{g}\right)(x) = \frac{x + 3}{x + 2}$, for $x \ne -2, 3$

A company's revenue for selling $x$ items is $R(x) = -0.5x^2 + 40x$ dollars and its cost is $C(x) = 10x + 250$. Find the profit function, the break-even quantities and the maximum profit.

$P(x) = -0.5x^2 + 30x - 250$; break even at $10$ and $50$ items; maximum profit $200$ dollars at $30$ items

Grade 12 · MHF4U · Advanced Functions

MHF4U: Sums, Products and Quotients of Functions

Two functions can be combined into a new one by doing arithmetic on their outputs. $(f + g)(x)$ means $f(x) + g(x)$ : for each $x$ , add the two $y$ -values. Subtraction, multiplication and division work the same way, and the only real care needed is with the domain.

How It Works

The four operations. $(f + g)(x) = f(x) + g(x)$ , $(f - g)(x) = f(x) - g(x)$ , $(fg)(x) = f(x) g(x)$ and $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$ . With equations, substitute and simplify. To evaluate at one number, you can also find $f(2)$ and $g(2)$ separately and combine the two numbers.

The domain. A combination is only defined where both $f$ and $g$ are, so its domain is the overlap of the two domains. A quotient also leaves out every $x$ where $g(x) = 0$ . Find the domain from the original functions, before you simplify: $\frac{x^2 - 9}{x^2 - x - 6}$ simplifies to $\frac{x + 3}{x + 2}$ , but $x = 3$ is still excluded.

Ordered pairs and graphs. When $f$ and $g$ are sets of ordered pairs, keep only the $x$ -values that appear in both, and combine their $y$ -values. On a graph, add or subtract the heights of the two curves at the same $x$ . Where one graph crosses the $x$ -axis, the sum equals the other function, which makes a useful check.

Applications. Profit is revenue minus cost, $P(x) = R(x) - C(x)$ , where $x$ is the number of items sold. Break-even is where $P(x) = 0$ , and the maximum profit of a quadratic profit function is at its vertex.

Even and odd. The product or quotient of two even functions, or of two odd functions, is even; an even times an odd is odd. A sum of an even and an odd function is usually neither.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
For $f(x) = 3x - 2$ and $g(x) = x^2 + 1$ , find $(f + g)(x)$ and $(f - g)(2)$ .
Show solution
1. $(f + g)(x) = 3x - 2 + x^2 + 1 = x^2 + 3x - 1$ .
2. $f(2) = 4$ and $g(2) = 5$ , so $(f - g)(2) = 4 - 5 = -1$ .
Answer: $(f + g)(x) = x^2 + 3x - 1$ ; $(f - g)(2) = -1$
Easy
$f = \{(-1, 4), (0, 2), (2, 5), (3, 1)\}$ and $g = \{(-2, 3), (0, -1), (2, 4), (3, 6)\}$ . Find $f + g$ and $fg$ .
Show solution
1. Only $x = 0, 2, 3$ appear in both, so $-1$ and $-2$ drop out.
2. Add the $y$ -values: $(0, 2 - 1)$ , $(2, 5 + 4)$ , $(3, 1 + 6)$ .
3. Multiply them: $(0, 2 \cdot (-1))$ , $(2, 5 \cdot 4)$ , $(3, 1 \cdot 6)$ .
Answer: $f + g = \{(0, 1), (2, 9), (3, 7)\}$ ; $fg = \{(0, -2), (2, 20), (3, 6)\}$
Medium
For $f(x) = \sqrt{x - 1}$ and $g(x) = \sqrt{5 - x}$ , find the domain of $f + g$ and of $fg$ , and evaluate $(fg)(3)$ .
Show solution
1. $f$ needs $x \ge 1$ and $g$ needs $x \le 5$ . Both hold for $1 \le x \le 5$ .
2. That overlap is the domain of $f + g$ and of $fg$ .
3. $(fg)(3) = \sqrt{2} \cdot \sqrt{2} = 2$ .
Answer: Both have domain $1 \le x \le 5$ ; $(fg)(3) = 2$
Medium
For $f(x) = x^2 - 9$ and $g(x) = x^2 - x - 6$ , simplify $\left(\frac{f}{g}\right)(x)$ and state its domain.
Show solution
1. Factor: $\frac{(x - 3)(x + 3)}{(x - 3)(x + 2)}$ .
2. $g(x) = 0$ at $x = 3$ and $x = -2$ , so both are excluded.
3. Cancel the common factor: $\frac{x + 3}{x + 2}$ , still with $x \ne 3$ .
Answer: $\left(\frac{f}{g}\right)(x) = \frac{x + 3}{x + 2}$ , for $x \ne -2, 3$
Test level
A company's revenue for selling $x$ items is $R(x) = -0.5x^2 + 40x$ dollars and its cost is $C(x) = 10x + 250$ . Find the profit function, the break-even quantities and the maximum profit.
Show solution
1. $P(x) = R(x) - C(x) = -0.5x^2 + 40x - 10x - 250 = -0.5x^2 + 30x - 250$ .
2. Break even: $-0.5x^2 + 30x - 250 = 0$ . Multiply by $-2$ : $x^2 - 60x + 500 = 0$ , so $(x - 10)(x - 50) = 0$ .
3. The vertex is midway between the zeros, at $x = 30$ : $P(30) = -450 + 900 - 250 = 200$ .
Answer: $P(x) = -0.5x^2 + 30x - 250$ ; break even at $10$ and $50$ items; maximum profit $200$ dollars at $30$ items

Learn It on Martian Lab

Martian Lab teaches this in the Combining Functions lesson of its MHF4U Combining Functions unit. Watch it, then take the Combining Functions practice tests to see if it stuck.

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More Combining Functions topics: Composition of Functions · Average vs Instantaneous Rate of Change

More MHF4U topics: Long and Synthetic Division · Vertical and Horizontal Asymptotes · Proving Trig Identities · Solving Trig Equations · Laws of Logarithms · Solving Logarithmic Equations · End Behaviour and Degree · Finding the Equation of a Polynomial · Even and Odd Functions · Remainder and Factor Theorem · Solving Polynomial Equations · Polynomial Inequalities · Sum and Difference of Cubes · Holes in Rational Functions · Graphing Rational Functions · Rational Equations and Inequalities · Radian Measure · Unit Circle and Special Angles · Compound Angle Formulas · Double Angle Formulas · Graphing Sine and Cosine with Transformations · Graphs of Tan, Sec, Csc and Cot · Sinusoidal Word Problems · Solving Exponential Equations · Graphing Logarithmic Functions · Exponential and Log Word Problems

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