Combined functions

Combined functions involve the operations of addition, subtraction, multiplication and division applied to two or more functions. A hybrid function is also an example of combined functions.

Combined function is a key part of understanding how functions interact and combine to form new expressions, making it a foundation for further studies in algebra, calculus and beyond. In particular, the domain and range of the resulting functions will be affected by these combinations which depends on the individual functions and the operation used.

Combine functions using algebraic operations

One way to combine functions is to use substitution which was introduced in the previous topic, composite functions. Another way to combine functions is using algebraic operations such as addition, subtraction, multiplication or division. When combining functions this way, the function outputs are combined. Thus, when adding function outputs, just add the entire functions together. The same for the other operations.

Combine Functions with Algebraic Operations

Suppose \(f\) and \(g\) are functions and \(x\) is in both the domain of \(f\) and the domain of \(g\).

  • The sum of \(f\) and \(g\), denoted \(f + g\), is the function defined by the formula

    • \[(f +g)(x) = f(x) + g(x)\]

  • The difference of \(f\) and \(g\), denoted \(f - g\), is the function defined by the formula

    • \[(f - g)(x) = f(x) - g(x)\]

  • The product of \(f\) and \(g\), denoted \(f \times g\), is the function defined by the formula

    • \[(f \times g)(x) = f(x) \times g(x)\]

  • The quotient of \(f\) and \(g\), denoted \(\frac{f}{g}\), is the function defined by the formula

    • \[\left(\frac{f}{g}\right)\left(x\right) = \frac{f(x)}{g(x)} \text{where } g(x) \neq 0\]

Worked Example

Given \(f\left(x\right) = x^2 - 1\) and \(g\left(x\right) = x + 1\), find

  1. \(\left(f + g\right)\left(x\right)\)
  2. \(\left(f - g\right)\left(x\right)\)
  3. \(\left(f \times g\right)\left(x\right)\)
  4. \(\left(\frac{f}{g}\right)\left(x\right)\)

Solution

  1. \(\left(f + g\right)\left(x\right) = f\left(x\right) + g\left(x\right) = x^{2} - 1 + x + 1 = x^{2} + x\)
  2. \(\left(f - g\right)\left(x\right) = f\left(x\right) - g\left(x\right) = x^2 - 1 - \left(x + 1\right) = x^2 - x - 2\)
  3. \(\left(f \times g\right)\left(x\right) = f\left(x\right) \times g\left(x\right) = \left(x^2 - 1\right) \times\left(x + 1\right) = x^3 + x^2 - x -1\)
  4. \(\left(\frac{f}{g}\right)\left(x\right) = \frac{f(x)}{g(x)} = \frac{x^2 - 1}{x + 1} = \frac{(x + 1)(x - 1)}{(x + 1)} = x - 1\)

Determining the domain and range of combined functions

After functions are combined using algebraic operations, domains and ranges of these resulting functions will also be affected. The domain of the combined function is the intersection of the two domains of two functions, and the range is determined in terms of the domain.

A 4-step process could be followed to determine the domain and range of combined functions.
Step 1. Simplify the combined functions (factorise where possible)
Step 2. Determine the domain of each of the two or more functions being combined
Step 3. The domain of the combined function consists of those values of \(x\) which belong to the domain of both functions from Step 2 such that the combined function is defined.
Note: If the combined function is formed by dividing two functions, then make sure to remove any \(x\) values which makes the denominator zero
Step 4. Determine the range of the combined function in terms of the domain (found in Step 3)

Worked Example

Given \(f\left(x\right) = x^2 - 2x\) and \(g\left(x\right) = -2x + 3\) , find \(h\left(x\right) = (f - g)(x)\) and determine the domain and range of \(h(x)\).

Solution

Step 1. Simplify the combined function (factorise where possible)
\[h\left(x\right) = \left(f - g\right)\left(x\right) =x^2 - 2x - \left(-2x + 3\right) = x^2 - 2x + 2x - 3 = x^2 -3 \]
Step 2. Determine the domain of each of the two functions being combined
The domain of \(f\left(x\right)\) is \(\mathbb{R}\), and the domain of \(g\left(x\right)\) is also \(\mathbb{R}\).
Step 3. The domain of the combined function consists of those values of \(x\) which belong to the domain of both functions from Step 2 such that the combined function is defined.
From Step 2, domain of both functions \(f\left(x\right)\) and \(g\left(x\right)\) is \(\mathbb{R}\). Moreover, there are no values of \(x\) for which the function \(h\left(x\right)\) is undefined. Therefore, the domain of the combined function \(h\left(x\right)\) is \(\mathbb{R}\).
Step 4. Determine the range of the combined function in terms of the domain (found in Step 3)
Since \(h\left(x\right) = x^2 - 3\) and \(x^2 \geq 0\), the smallest value of \(h\left(x\right)\) is -3 which occurs when \(x = 0\). Therefore, the range of \(h\left(x\right)\) is \(\left[-3,\infty\right)\).


Worked Example

Given \(f\left(x\right) = {-2x}^2 + 7x - 6\) and \(g\left(x\right) = x - 2\), find \(h\left(x\right) = \left(\frac{f}{g}\right) (x)\) and determine the domain and range of \(h(x)\).

Solution

Step 1. Simplify the combined function (factorise where possible)

\[h(x) = \left(\frac{f}{g}\right)\left(x\right) = \frac{f\left(x\right)}{g\left(x\right)} = \frac{-2x^{2} + 7x -6}{x - 2} = \frac{\left(-2x + 3\right)\left(x - 2\right)}{x - 2} = -2x + 3\]

Step 2. Determine the domain of each of the two functions being combined

The domain of \(f\left(x\right)\) is \(\mathbb{R}\), and and the domain of \(g\left(x\right)\) is also \(\mathbb{R}\)

Step 3. The domain of the combined function consists of those values of \(x\) which belong to the domain of both functions from Step 2 such that the combined function is defined.

Note: In this case, the denominator cannot be zero so that \(x - 2 \neq 0, x \neq 2\).

Therefore, the domain of \(h(x)\) is all real numbers except 2 which is written as \(\left(-\infty,2\right) \cup \left(2,\infty\right)\).

Step 4. Determine the range of the combined function in terms of the domain (found in Step 3)

The range of the combined function h(x) can be found by determining the possible values of \(h\left(x\right) = -2x + 3\) that are obtained when plugging in any \(x\) value that is not equal to 2.
The graph of \(h(x)\) is a straight-line graph which has a negative slope of -2, but since \(x \neq 2, h\left(2\right) = -1\), so -1 is not in the range of \(h(x)\). Therefore, the range of \(h(x)\) is all real numbers except -1 which is written as \(\left(-\infty,-1\right) \cup \left(-1,\infty\right)\).

Hybrid (Piecewise) Functions

A hybrid function (also known as piecewise function) is a function that contains multiple rules across different subsets of the domain.
Hybrid functions are represented using the following notation:

\[
h(x) =
\left\{
\begin{array}{ll}
f(x), & a \leq x \leq b \\
g(x), & b < x \leq c \\
&\vdots &\vdots
\end{array}
\right\}
\]

For example,

Worked Example

\[h(x) = \left\{\begin{array}{ll} x^2 + 2, x \leq 1 \\ 3x, x > 1\end{array}\right\}\]

This hybrid function \(h(x)\) is composed of a quadratic function \(x^2 + 2\) only when \(x\) is less than or equal to 1 and a linear function \(3x\) when \(x\) is greater than 1.

  • Graph the quadratic function \(x^2 + 2\) first, the turning point is at \((0,2)\) which produces the minimum \(y\) value
  • Only graph the parabola for \(x \le 1\) which is the restriction of the domain. Therefore, the end point of \(y  = x^2 + 2\) is at \(y = 1 + 2 = 3\). The coordinates of the end point are \(\left(1,3\right)\)
  • Then sketch the graph of \(3x\) which is a straight line. \((1,3)\) is lying on the straight line which means that it is connected with the end point of the parabola