Operations with Functions
Functions can be added, subtracted, multiplied and divided just like numbers! Functions can be applied in a variety of ways to describe all sorts of complicated relationships. These relationships can be expressed using various types of math operations such as addition, subtraction, multiplication, and division to combine 2 or more different functions.
These operations can be expressed in the following table:
| Operation | Formula |
|---|---|
| Addition | \(h(x) = f(x) + g(x)\) |
| Subtraction | \(h(x) = f(x) - g(x)\) |
| Multiplication | \(h(x) = f(x) \cdot g(x)\) |
| Division | \(h(x) = \cfrac{f(x)}{g(x)}\) |
The Superposition Principle states that the sum of 2 or more functions can be determined by adding the \(y\)-coordinates at each point along the \(x\)-axis. This principle can also be applied to determine the difference of 2 or more functions by subtracting the respective \(y\)-coordinates.
Additionally, the Superposition Principle can be used to determine the domain of the combined function.
Example
Algebraically determine a combined equation, \(h(x)\), using the functions \(f(x) = x^2\) and \(g(x)=5\). We can then graph the combined function and determine its domain and range.
First, we can determine \(h(x)\) by adding \(f(x)\) and \(g(x)\):
\(h(x) = f(x) + g(x)\)
\(h(x) = x^2 + 5\)
Next, we can graph \(h(x)\) using the Superposition Principle. To do so, we can first graph \(f(x)\) and \(g(x)\) on the same set of axes:
We can now graph the combined function. We can also create a table of values to determine the \(y\)-coordinates:
| \(x\) | \(f(x) = x^2\) | \(g(x) = 5\) | \(h(x) = x^2 + 5\) |
|---|---|---|---|
| \(-2\) | \(4\) | \(5\) | \(9\) |
| \(-1\) | \(1\) | \(5\) | \(6\) |
| \(0\) | \(0\) | \(5\) | \(5\) |
| \(1\) | \(1\) | \(5\) | \(6\) |
| \(2\) | \(4\) | \(5\) | \(9\) |
From this graph, we can determine that the domain is \(\boldsymbol{\{x \in \mathbb{R} \}}\) and the range is \(\boldsymbol{\{y \in \mathbb{R} | y \ge 3 \}}\).
Add \(p(x) = 2x^2+7x\) and \(q(x) = 3x^2 + 10\)
We can determine \(h(x)\) by adding \(p(x)\) and \(q(x)\):
\(h(x) = p(x) + q(x)\)
\(h(x) = 2x^2+7x + 3x^2 + 10\)
\(h(x) = 5x^2 + 7x + 10\)
Therefore, we can determine that \(\boldsymbol{h(x) = 5x^2 + 7x + 10}\).
Subtract \(m(x) = 4x^3 + x\) by \(n(x) = 2x - 8\)
We can determine \(h(x)\) by subtracting \(p(x)\) and \(q(x)\):
\(h(x) = m(x) - n(x)\)
\(h(x) = 4x^3 + x - (2x - 8)\)
\(h(x) = 4x^3 + x -2x + 8\)
\(h(x) = 4x^3 -x + 8\)
Therefore, we can determine that \(\boldsymbol{h(x) = 4x^3 -x + 8}\).
Multiply \(u(x) = x + 5\) by \(v(x) = 2x - 3\)
We can determine \(h(x)\) by multiplying \(u(x)\) and \(v(x)\):
\(h(x) = u(x) \cdot v(x)\)
\(h(x) = (x + 5)(2x - 3)\)
\(h(x) = 2x^2 -3x + 10x -15\)
\(h(x) = 2x^2 +7x -15\)
Therefore, we can determine that \(\boldsymbol{h(x) = 2x^2 +7x -15}\).
Divide \(i(x) = x^2 + 5x + 6\) by \(j(x) = x + 2\)
We can determine \(h(x)\) by dividing \(i(x)\) by \(j(x)\):
\(h(x) = \cfrac{i(x)}{j(x)}\)
\(h(x) = \cfrac{x^2 + 5x + 6}{x + 2}\)
\(h(x) = \cfrac{(\cancel{x+2})(x+3)}{\cancel{x+2}}\)
\(h(x) = x+3\)
Therefore, we can determine that \(\boldsymbol{h(x) = x+3}\).