10-9 Combining Functions

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 $\frac{7}{1} = 7$.

Property

When finding the difference of two functions, $(f - g)(x)$, you must substitute the entire expression for $g(x)$ inside parentheses. The negative sign must then be distributed across every single term of $g(x)$ before combining like terms.

Examples

• Distributing the Negative: Let $f(x) = 3x^2 + 5x + 1$ and $g(x) = x^2 - 4x + 6$. Find $(f - g)(x)$.

Setup with parentheses: $3x^2 + 5x + 1 - (x^2 - 4x + 6)$
Distribute the negative (flip all signs in $g$): $3x^2 + 5x + 1 - x^2 + 4x - 6$
Combine like terms: $2x^2 + 9x - 5$

• The Common Error: A student writes $3x^2 + 5x + 1 - x^2 - 4x + 6$.

This is incorrect. The student only subtracted the first term of $g(x)$ and forgot to flip the signs of $-4x$ and $6$.

Explanation

Property

To find the product of two functions, $(fg)(x)$, multiply their polynomial expressions using the Distributive Property (or the vertical multiplication method).

When you multiply two non-zero polynomials, the degree (highest exponent) of the new combined function is exactly the sum of the degrees of the original functions.

Examples

• Binomial times Trinomial: Let $f(x) = x + 2$ and $g(x) = x^2 + 3x + 1$. Find $(fg)(x)$.

Distribute the $x$: $x(x^2 + 3x + 1) = x^3 + 3x^2 + x$
Distribute the 2: $2(x^2 + 3x + 1) = 2x^2 + 6x + 2$
Combine like terms: $x^3 + 5x^2 + 7x + 2$

• Predicting the Degree: Let $P(x) = 3x^5 + x$ (degree is 5) and $Q(x) = 2x^2 - 1$ (degree is 2).

Without doing the full multiplication, we know the degree of the product $(PQ)(x)$ will be $5 + 2 = 7$.

Explanation