1.15: Operations on Functions
Just as numbers can be added, subtracted, multiplied, and divided, so too can functions. Combining functions in this way can often have surprising results, as the resultant function may not have a graph that appears similar to that of either input function's graph.
How can you tell, before completing the entire operation and graphing the result, whether the new function is likely to resemble one of the input functions? How do you describe combined functions without a graph?
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James Sousa  The Algebra of Functions
Guidance
Sums and Differences of Functions
Consider the function: f(x) = 48/x + 2x^{2}.
Notice that the equation has two terms:
 The first term: 48/x
 The second term: 2x^{2}
Therefore we can think of the function f(x) as the sum of two other functions:
 The reciprocal function g(x) = 48/x
 The quadratic function b(x) = 2x^{2}
When we add the functions together, we get a new type of graph that resembles both the graphs of g(x) and b(x):
The graph on the right is f(x). The right portion of f(x) resembles the parabola b(x), but is asymptotic to the yaxis. The left portion of f(x) resembles the left side of g(x), as both functions are asymptotic to the negative yaxis.
There are two points to be stressed here: first, that we can add functions together, and second, that the resulting sum may be a different kind of function from the original two.
The sum or difference of a function is more likely to resemble the original two functions if they are from the same family.
For example, if two functions from the linear function family are added together, the sum function is also a member of the linear family.
Example A
If f(x) = x^{3} + 2x^{2} and g(x) = x^{2}  5, what is f  g? What does the graph look like?
Solution:
The difference is: f  g = x^{3} + 2x^{2}  (x^{2}  5) = x^{3} + 2x^{2}  x^{2} + 5 = x^{3} + x^{2} + 5
The graph of the new function, along with f(x), is shown here:
Because f(x) and the new function y = x^{3} + x^{2} + 5 are both members of the cubic family, they have similar shapes.
To recap: When we add or subtract functions, the resulting sum or difference function may be in the same family as one or both of the original functions, or it may be a different type of function. The resultant function is more likely to be in the same family if both of the initial functions are in the same family as each other.
Example B
Given f(x) = 2x^{2} and g(x) = x + 1, find r(x) = f(x)/ g(x) and t(x) = g(x)/f(x).
Solution:
r(x) = f(x)/g(x) = 2x^{2}/(x + 1). This is a rational function, and does not have a horizontal asymptote. It does, however, have a vertical asymptote at x = 1, as the domain excludes x = 1.
t(x) = g(x)/ f(x) = (x + 1)/2x^{2}. This is also a rational function. This function has a horizontal asymptote at y = 0 (the xaxis), and a vertical asymptote at x = 0 (the yaxis).
Notice that the graph of this function crosses its asymptote at (1, 0), but then as x approaches \begin{align*}\infty\end{align*}, the function values approach 0.
In general, if we multiply linear and polynomial functions (quadratics, cubics, and other such functions with higher exponents, such as y = x^{4} + 3x^{2} + 2), we will obtain other polynomial functions. If we divide these kinds of functions, we will obtain other polynomial functions, or rational functions.
Multiplying and dividing other types of functions may result in more complicated graphs.
Example C
Consider the functions f(x) = x and g(x) = x^{2}  4. Identify the graphs of f(x)/g(x) = x/(x^{2}  4) and g(x)/f(x) = (x^{2}  4)/ x.
Solution:
The graphs of these two functions are not unlike the rational functions discussed in a later lesson.
Did you discover the trick for identifying when a resultant function graph is likely to resemble the input graphs, as mentioned at the beginning of the lesson? The sum or difference of a function is more likely to resemble the original two functions if they are from the same family. In other words, if you are adding or subtracting two quadratic equations, the result is likely to be quadratic, and have a similar graph. 

Vocabulary
Function sum: The result of the addition of two functions.
Function difference: The result of the subtraction of two functions.
Asymptote: A line on a graph toward which the output of a given function may approach, but never quite reach.
Guided Practice
Questions

1) Given \begin{align*}f(x) = 4x^2  7\end{align*} and \begin{align*}g(x) = 3x^2 2x + 8\end{align*}:
 Find and graph (use technology): \begin{align*}(f + g)(x)\end{align*}

2) Multiply the function by the scalar value
 If \begin{align*}f(x) = 3x + 10\end{align*} find \begin{align*}3 \cdot f(x)\end{align*}

3) Given \begin{align*}f(x) = 3x  7\end{align*} and \begin{align*}g(x) = 4x + 6\end{align*}:
 Find and graph (use technology) \begin{align*}(f \cdot g)(x)\end{align*}
Solutions

1) Step 1: Recall that \begin{align*}(f + g)(x) = f(x) + g(x)\end{align*}
 Step 2: Substitute \begin{align*}f(x) + g(x) = (4x^2  7) + (3x^2  2x + 8)\end{align*}
 Step 3: Combine like terms \begin{align*}(f + g)(x) =7x^2  2x + 1\end{align*}
 So our answer is: \begin{align*}(f + g)(x) = 7x^2  2x + 1\end{align*}

 The graph of \begin{align*}f(x) = 7x^2  2x +1\end{align*} looks like:


2) To multiply a function by a scalar, multiply each term of the function by the scalar:
 Step 1: Substitute: \begin{align*}3f(x) = 3(3x + 10)\end{align*}
 Step 2: Distribute: \begin{align*}3f(x) = 9x + 30\end{align*}
 So our answer is: \begin{align*}3f(x) = 9x + 30\end{align*}

3) Step 1: Recall that \begin{align*}(f \cdot g)(x) = f(x) \cdot g(x)\end{align*}
 Step 2: Substitute: \begin{align*}f(x) \cdot g(x) = (3x  7)(4x + 6)\end{align*}
 Step 3: Distribute (FOIL): \begin{align*}(f \cdot g)(x) = 12x^2 + 18x  28x  42\end{align*}
 Step 4: Combine like terms: \begin{align*}(f \cdot g)(x) = 12x^2  10x  42\end{align*}
 So our answer is:\begin{align*}(f \cdot g)(x) = 12x^2  10x  42\end{align*}

 The graph of \begin{align*}f(x) = 12x^2  10x  42\end{align*} looks like this:

Practice
Given \begin{align*}f(x) = \frac{x^3}{x + 1}\end{align*} and \begin{align*}g(x) = x(x + 1)\end{align*} find each of the following:
 \begin{align*}(fg)(x) = \end{align*}
 \begin{align*}(fg)(1) = \end{align*}
 \begin{align*} (\frac{f}{g})(x)=\end{align*}
Simplify the following:
 If \begin{align*}f(x) = 2x + 4\end{align*}and \begin{align*}g(x) = 3x  7\end{align*}, find \begin{align*}(f + g(x))\end{align*}.
 If \begin{align*}g(x) = \frac{2}{3}x + 12\end{align*} and \begin{align*} h(x) = \frac{1}{4}x + 7\end{align*}, find\begin{align*}(g + h)(x)\end{align*}
 If \begin{align*}f(x) = 4x^2  10\end{align*} and \begin{align*}g(x) =5x^2  2x  3\end{align*}, find\begin{align*}(f + g)(x)\end{align*}
 If \begin{align*}f(x) = 6x^2  3x + 5\end{align*} and \begin{align*}g(x) = 4x^2 + 5x  8\end{align*}, find\begin{align*}(g  f)(x)\end{align*}.
 If\begin{align*}g(x) = 6x  8\end{align*}, find \begin{align*}\frac{3}{2}g (x)\end{align*}.
 If \begin{align*}g(x) = 2x^2 + 3\end{align*} and \begin{align*}h(x) = 3x  6\end{align*}, find\begin{align*}(g \cdot h)(x)\end{align*}.
Evaluate and Graph:
 if\begin{align*}f(x) = 6x + 4\end{align*} and \begin{align*}g(x) = 7x  8\end{align*}, find\begin{align*}(f + g) (3)\end{align*}.
 If\begin{align*}f(x) \frac{1}{4}x + 3\end{align*} and \begin{align*}h(x) = \frac{3}{2}x + 6\end{align*}, find \begin{align*}(g + h)(12)\end{align*}.
 If\begin{align*} g(x) = 5x^2  4x + 3\end{align*}and\begin{align*}h(x) = 2x  7\end{align*}, find\begin{align*}(g  h) (2)\end{align*}.
 If\begin{align*}g(x) = 4x^3  3x\end{align*} find \begin{align*}5g(6)\end{align*}.
 If\begin{align*}h(x) = 4x  7\end{align*} find \begin{align*}2h(5)\end{align*}.
 If\begin{align*}f(x) = x + 4\end{align*} and \begin{align*}g(x) = 3x  6\end{align*}, find \begin{align*}(f \cdot g)(1)\end{align*}.
 If\begin{align*}h(x) = x^4\end{align*} and\begin{align*}g(x) = x  12\end{align*}, find\begin{align*}(h \cdot g)(2)\end{align*}
Try these more challenging problems.
 Solve and graph.
 If \begin{align*}f(x) = 4x  7\end{align*}, \begin{align*}g(x) = 3x + 18\end{align*}, and \begin{align*}h(x) = 5x +2\end{align*}, find\begin{align*}(f + g h)(x)\end{align*}.
 If \begin{align*}f(x) = 6x  8\end{align*},\begin{align*}y(x)  \frac{1}{2}x\end{align*}, and \begin{align*}h(x) = x + 4\end{align*}, find \begin{align*}(f \cdot g \cdot h) (x)\end{align*}
 If \begin{align*}g(x) = 3x  7\end{align*} and \begin{align*}(g \cdot h)(x) = 15x^2  47x + 28\end{align*}, find\begin{align*}(h)(x)\end{align*}.
Asymptotes
An asymptote is a line on the graph of a function representing a value toward which the function may approach, but does not reach (with certain exceptions).Difference
The result of a subtraction operation is called a difference.Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .Horizontal Asymptote
A horizontal asymptote is a horizontal line that indicates where a function flattens out as the independent variable gets very large or very small. A function may touch or pass through a horizontal asymptote.Sum
The sum is the result after two or more amounts have been added together.Vertical Asymptote
A vertical asymptote is a vertical line marking a specific value toward which the graph of a function may approach, but will never reach.Image Attributions
Here you will learn how to perform the standard mathematical operations of addition, subtraction, multiplication, and division on functions. You will also explore the graphs that result from these operations.
Concept Nodes:
Asymptotes
An asymptote is a line on the graph of a function representing a value toward which the function may approach, but does not reach (with certain exceptions).Difference
The result of a subtraction operation is called a difference.Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .Horizontal Asymptote
A horizontal asymptote is a horizontal line that indicates where a function flattens out as the independent variable gets very large or very small. A function may touch or pass through a horizontal asymptote.Sum
The sum is the result after two or more amounts have been added together.Vertical Asymptote
A vertical asymptote is a vertical line marking a specific value toward which the graph of a function may approach, but will never reach.