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?

### Operations on Functions

#### Sums and Differences of Functions

Consider the function: *f*(*x*) = 48/*x* + 2*x*^{2}.

Notice that the equation has two terms:

The first term: 48/*x*

The second term: 2*x*^{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*) = 2*x*^{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 *y*-axis. The left portion of *f*(*x*) resembles the left side of *g*(*x*), as both functions are asymptotic to the negative *y*-axis.

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.

### Examples

#### Example 1

Earlier, you were asked if you could discover the trick for identifying when a resultant function graph is likely to resemble the input graphs.

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.

#### Example 2

If *f*(*x*) = *x*^{3} + 2*x*^{2} and *g*(*x*) = *x*^{2} - 5, what is *f* - *g*? What does the graph look like?

The difference is: *f* - *g* = *x*^{3} + 2*x*^{2} - (*x*^{2} - 5) = *x*^{3} + 2*x*^{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 3

Given *f*(*x*) = 2*x*^{2} and *g*(*x*) = *x* + 1, find *r*(*x*) = *f*(*x*)/ *g*(*x*) and *t*(*x*) = *g*(*x*)/*f*(*x*).

*r*(*x*) = *f*(*x*)/*g*(*x*) = 2*x*^{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 *x*-axis), and a vertical asymptote at *x* = 0 (the *y*-axis).

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} + 3*x*^{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 4

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*}.

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:

#### Example 5

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*}.

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*}

#### Example 6

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*}.

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:

### Review

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*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.15.