4.05 Function operations
Concept summary
Just as we can perform operations on polynomials, we can also perform operations on different functions - adding, subtracting, multiplying or dividing them - provided we follow specific rules.
Operations with functions are defined using special notation:
- Sum: \left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)
- Difference: \left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)
- Product: \left(f\cdot g\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)
- Quotient: \left(\dfrac{f}{g}\right)\left(x\right)=\dfrac{f\left(x\right)}{g\left(x\right)} \text{, where } g\left(x\right)\neq 0
With each operation, the domain of the new function becomes the intersection or overlap of the domains of the original functions. The exception is that in the case of a quotient function, the new function's domain is further restricted to exclude values that make the denominator function zero.
Worked examples
Example 1
Consider the following pair of functions:
\begin{aligned} f\left(x\right) & = -5x+5\\\ g\left(x\right) & = 2x^2+3x-10 \end{aligned}
Find \left(f+g\right)\left(x\right)
Approach
Solution
Find \left(f-g\right)\left(x\right)
Approach
Solution
Find \left(f \cdot g\right)\left(x\right)
Approach
Solution
Example 2
The table shows some of the outputs of the functions f\left(x\right) and g\left(x\right).
Use the table to evaluate the following:
| x | f\left(x\right) | g\left(x\right) |
|---|---|---|
| 0 | -2 | 8 |
| 1 | 5 | 7 |
| 2 | 12 | 4 |
| 3 | 19 | -1 |
| 4 | 26 | -8 |
| 5 | 33 | -17 |
\left(f + g\right)\left(4\right)
Approach
Solution
\left(f\cdot g\right)\left(3\right)
Approach
Solution
\left(\dfrac{ f}{g}\right)\left(2\right)