Our two main tools for investigating functions are formulas and graphs. Not every function has a formula, but when it does there is a connection between the formula and graph.

Domain

When a function is described with a formula, then the domain is described in writing - usually some sort of set notation. Our favorite way is through interval notation. Set builder notation is also used. Sometimes the domain is just described with words.

All of the domain numbers are pictured as lying on the horizontal axis in a graph. They are not plotted as points on the horizontal axis, unless the function value just happens to equal \(0\) at a domain number.

To aid a discussion, the horizontal axis might be shaded to illustrate the domain.

Range

The same idea goes for the range of a function, except these values are imagined along the vertical axis.

Pairs

The pairs are the most important part of a function. They give the connection between the domain and range. Formulas do not explicitly give pairs. You can assemble an individual pair, one-at-a-time, by evaluating the formula at a particular domain number.

Graphs display pairs. The dots included in the graph are visually encoding the function pairs. Their coordinates can be deciphered into a domain number and function value. The domain number is the first coordinate and the function value is the second coordinate.

Linear functions are functions that can be described by formulas that look like \(L(x) = A \cdot x + B\), with \(A\) and \(B\) both real numbers. ( When \(A = 0\) ), the linear function is called a constant function. Graphs of linear functions are lines. It takes two points to draw a line. Given a formula, we can evaluate it twice, from which we can create two points, plot them, and draw the line.

Let \(L(x) = 2x-3\) with its domain and range both \((-\infty , \infty )\).

• \(L(-2) = 2(-2) - 3 = -7\), which gives the point \(\left (\answer {-2}, \answer {-7}\right )\).
• \(L(4) = 2(4) - 3 = 5\), which gives the point \((4, 5)\).

The arrows indicate that the graph continues with the same pattern - illustrating that the domain is \((-\infty , \infty )\) as well as the range.

Preview: Rate of Change

A Peek Ahead

Much of this course will be spent investigating rate of change.

Linear functions have a constant growth rate or rate of change. Whenever the domain changes by a given amount, then the function changes by a constant multiple of that domain change.

Linear functions have a constant growth rate.

Later in this course, we’ll compare this to a constant percentage growth rate, which characterizes exponential functions. Whenever the domain changes by a given amount, then the function changes by a constant percentage of the current function value. A constant percentage growth rate results in a formula of the form \(E(t) = a \cdot r^t\), where \(a\) is a nonzero real number and \(0 < r\).

Let \(E(t) = 2 \cdot 1.25^t\) with a domain consisting of all real numbers, \(\mathbb {R}\).

When \(t\) changes from \(a\) to \(a+1\), \(E\) grows by \(25\%\) of \(E(a)\). Whenever \(t\) changes by \(1\), \(E\) grows by \(25\%\), because the function value is multiplied by another \(1.25 = 1 + 0.25\).

Since \(1.25 > 1\), whenever the domain number, \(t\), increases, so does \(E(t)\). \(E\) is an increasing function. Its graph goes uphill to the right.

In addition, \(1.25\) to any power will be positive. This together with \(2>0\) tells us that \(E(t) > 0\) for all \(t\).

Note: If we change \(r\) to be \(0 < r < 1\) (keeping \(a>0\)), the exponential function, \(E(t) = a \cdot r^t\), becomes a decreasing function and its graph is reflected horizontally.

Let \(X(t) = 2 \cdot 0.8^t\) with a domain consisting of all real numbers, \(\mathbb {R}\).

Whenever \(t\) changes by \(1\), \(X\) decays by \(20\%\), because the function value is multiplied by another \(0.8 = 1 - 0.2\).

Since \(0.8 < 1\), whenever the domain number, \(t\), increases, \(X(t)\) decreases. \(X\) is a decreasing function. Its graph goes downhill to the right.

Still, \(0.8\) to any power will be positive. This together with \(2>0\) tells us that \(X(t) > 0\) for all \(t\).

The graph of an exponential function attempts to level off at a height of \(0\). This follows the idea that \(0.8\) raised to larger and larger powers will result in a smaller and smaller positive value, but never equal \(0\) itself.

Zeros

The real numbers experience a significant change in behavior at \(0\). The positive and negative numbers possess drastically different algebraic properties. \(0\) also sets itself aside with properties different from both the negative and positive real numbers.

For one thing, anything times \(0\) is \(0\).

In the other direction, we have the Zero Product Property.

This states that the ONLY way a product of two real numbers is \(0\) is if one of the numbers is \(0\).

For these reasons, we are interested in where functions have zero values.

Zeros of functions correspond to intercepts on the graph.

\(\blacktriangleright \) If \(a\) is a zero of the function \(f\), then \((a, f(a)) = (a, 0)\) is a point on the graph of \(f\).

\(\blacktriangleright \) If \((a, 0)\) is a point on the graph of \(f\), then \(f(a) = 0\) and \(a\) is a zero of \(f\).

Let \(P(w) = w^2 - 2w - 3\) with a domain consisting of all real numbers, \(\mathbb {R}\).

The graph has two intercepts.

It appears that \((-1,0)\) and \((3,0)\) are intercepts of the graph, which suggests that \(-1\) and \(3\) are zeros of \(P\). We can verify this via the formula.

• \(P(-1) = (-1)^2 -2(-1) - 3 = \answer {0}.\)
• \(P(3) = (3)^2 -2(3) - 3 = \answer {0}.\)

The arrows on the graph of \(P\) let us know that the graph keeps rising to the left and right, which means there are no other intercepts. \(P\) has two zeros.

Domain Types

We encounter functions in several ways, each affecting the domain of a function.

(a)

stated domain
A function may come already equipped with a stated domain. Graphs communicate a stated domain - just collect all of the first coordinates from the points. Many times we use interval notation to describe a stated the domain.

(b)

natural or implied domain
Mathematicians like shorthand. The best shorthand is just nothing. Nothing is used all over the place. If a function is described with a formula and there is no stated domain, then there is the natural or implied domain. The natural or implied domain is all real numbers that don’t cause a problem with the formula.

So far, we know of two problems: square (even) roots of negative numbers and fractions with \(0\) denominators. We will encounter a third problem, which will be logarithms of zero or negative numbers (later). Any real numbers that cause these problems are removed from the real numbers to obtain the natural or implied domain.

(c)

applied domain
We use functions to model many measuring situations. In such cases, we want our model to describe the situation. Therefore, the domain should not contain numbers that don’t fit the situation. An applied domain is a subset of the natural domain. The applied domain includes all of the real numbers that make sense in the application.

(d)

induced domain
We often create new functions from old functions. We have a laundry list of ways to accomplish this. One method is to keep the function structure the same and simply move the domain to another location. We’ll study these types of transformations extensively. For the moment, an example will give the idea.

Example: Let \(f(x)\) be a function with domain \([-2, 5]\). Define a new function by \(g(t) = f(t+4)\). To evaluate \(g\), you add \(4\) to \(g\)’s domain number and evaluate \(f\).

• The least number in the domain of \(f\) is \(-2\). Therefore, the least number in the domain of \(g\) must be \(-6\).
• The greatest number in the domain of \(f\) is \(5\). Therefore, the greatest number in the domain of \(g\) must be \(1\).

The domain of \(g\) is \([-6, 1]\) and it was induced (forced) from the domain of \(f\) by the defining equation.