Definition 6.5.
Suppose \(f\) is a real-valued function of a real variable. The graph of \(f\) is the graph of the equation \(y = f(x)\text{.}\) The graph consists of all points \((x,f(x))\text{,}\) with \(x\) in the domain of \(f\text{.}\)
Section 6.2 Graphing Functions
The graph of a function is a visual representation of the relationship between the inputs and corresponding output of a function. We typically plot inputs on the horizontal \(x\)-axis and the corresponding output on the vertical \(y\)-axis. Each input/output pair determines a point on the graph \((x,f(x))\text{.}\)
Example 6.6. The graph of the squaring function.
Let \(f(x) = x^2\text{,}\) the squaring function. The graph of \(f\) is the graph of the equation \(y=x^2\text{.}\) It consists of all points \((x,y) = (x,x^2)\text{,}\) where \(x\) is any real number. For instance, \(f(2) = 4\) so that \((2,4)\) is a point on the graph. Similarly, \(f(-3) = (-3)^2 = 9\) so that \((-3,9)\) is also point on the graph. The resulting curve is called a parabola.
Visualize the domain \((-\infty,+\infty)\) along the horizontal axis and the range \([0,+\infty)\) along the vertical axis. Observe how I have labeled the curve with its equation \(y=x^2\text{.}\) This is a practice should adhere to as well when graphing.
Example 6.7. A function given by its graph.
A function may be described entirely by its graph. For example, the odd looking graph below determines a function which I will call \(f\text{.}\) 
We can use the graph to evaluate \(f\text{.}\) For instance, \(f(-1)= 1\text{,}\) or at least appears to be so if the graph truly includes the point \((-1,1)\text{.}\) Similarly, \(f(0) \approx 1.5\text{,}\) which I’m definitely estimating here. This determines the \(y\)-intercept of the graph. A value of \(x\) where \(f(x) = 0\) is called a (real) zero of \(f\) and corresponds to an \(x\)-intercept of the graph. It appears that this function has zeros at \(x = -3\) and at \(x \approx 1.4\text{.}\)
The behavior of the graph at \(x=1\) is subtle. There is clearly a point plotted at \((1,1)\) so that \(f(1) = 1\text{.}\) The hole at \((1,2)\) is used to emphasize that no point is plotted here. In fact, if there was not a hole here as in 
then this would not be a the graph of a function as the value of \(x=1\) is associated to two different \(y\)-coordinates. This violates the uniqueness requirement in Definition 6.2
The domain of \(f\) can be visualized along the horizontal \(x\)-axis; the range along the vertical \(y\)-axis.
In this case, the domain of \(f\) is the interval \([-3,2)\text{.}\) We exclude \(2\) because no point is plotted at \(x=2\text{.}\) The range of \(f\) is the interval \((-2,2)\text{,}\) excluding both \(y = -2\) and \(y = 2\text{.}\)
A vertical line drawn to the left or the right of the circle does not intersect the graph at all. This is fine, as these correspond to values of \(x\) not in the domain of these functions.
When \(f\) is a function and we write \(y = f(x)\text{,}\) we are saying that \(y\) is a function of \(x\text{,}\) that is, the value of \(y\) is explicitly determined by the value of \(x\text{.}\) Graphically, this means a vertical line corresponding to a value of \(x\) in the domain of the function should intersect the graph once.
Principle 6.8. Vertical Line Test.
Suppose \(f\) is a real-valued function of the variable \(x\text{.}\) Then any vertical line will intersect the graph of \(y = f(x)\) at most once. In this case, the graph is said to pass the vertical line test.
On the other hand, if there is a vertical line which intersects a graph more than once, then this is not the graph of a function of \(x\text{.}\) In this case, the graph is said to fail the vertical line test.
Example 6.9. A circle is not the graph of any function.
Consider the graph of the equation \(x^2 + y^2 = 1\) which forms the unit circle of radius 1 centered at the origin. 
A vertical line drawn between \(-1 \lt x \lt 1\) intersects the circle exactly twice. The value of \(x\) here does not explicitly determine the value of \(y\text{.}\) Thus, \(y\) is not a function of \(x\) and the circle is not the graph of any function.
We can make some progress if we solve for \(y\) using the Square Root Principle 4.21:
\begin{gather*}
x^2 + y^2 = 1\\
y^2 = 1 - x^2\\
y = \pm \sqrt{1 - x^2}
\end{gather*}
There are really two equations here, \(y = \sqrt{1-x^2}\) and \(y = -\sqrt{1-x^2}\text{,}\) which correspond to the upper and lower semi-circles. These graphs do pass the vertical line test individually and so are graphs of functions.
