# Solved Examples on the Graphs of Functions

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(1.) Analyze the graph of the function based on increasing, decreasing, and constant intervals.
In other words: Based on the graph, at what intervals of x does the function, y increase. decrease, or remain constant?

$y \uparrow \;\;\;for\;\;\; x \in (-8, -2) \cup (0, 2) \cup (5, 7) \\[3ex] y \downarrow \;\;\;for\;\;\; x \in (-10, -8) \cup (-2, 0) \cup (2, 5)$
(2.) State whether the following statements are true or false.
(a.) Every graph represents a function.

(b.) The graph of a function y = ​f(x) always crosses the​ y-axis.

(c.) The point ​(​−7,​ −6) is on the graph of the equation x = 4y − 5.

(d.) The​ y-intercept of the graph of the function ​$y = f(x)$, whose domain is all real​ numbers, is $f(0)$.

(a.) This statement is false because of the Vertical Line Test.
The statement is false because a graph that crosses the​ y-axis two times does not represent a function.
If the vertical line intersects the graph at only one point, the graph is a function because no input, x has more than one output, y.
If the vertical line intersects the graph at more than point, the graph is not a function because there is at least an input, x that has more than one output, y.
The Vertical Line Test states that if a vertical line is drawn through the graph of a set of points in the rectangular coordinate system, the graph is a function if and only if the vertical line intersects the graph at only one point.
If a graph crosses the y-axis more than one time, it has failed the Vertical Line Test because it means that there is at least an input, x that has more than one output, y.

(b.) On the y-axis, the x-value is zero.
Some functions have domains that do not include zeros. For example, the parent function of a rational function is $y = \dfrac{1}{x}$
For that function, the graph will never cross the y-axis because the vertical asymptote, $x = 0$ is the y-axis
There are some graphs that do not cross the y-axis. Hence, te graph of a function does not always cross the y-axis
The statement is false.

$(c.) \\[3ex] x = 4y - 5 \\[3ex] (-7, -6) \implies \\[3ex] x = -7 \\[3ex] y = -6 \\[3ex] \implies \\[3ex] -7 \stackrel{?}{=} 4(-6) - 5 \\[3ex] -7 \stackrel{?}{=} -24 - 5 \\[3ex] -7 \ne -29 \\[3ex]$ The point ​(​−7,​ −6) is not on the graph of the equation x = 4y − 5.

(d.) The statement is true.
For all real number domain:
To find the y-intercept:
set x to 0 and
solve for y
Hence: for $y = f(x)$,
$y = f(0)$ is the value of the y-intercept.
But, please note that y-intercept is a point.
So, the y-intercept = $(0, y-value)$

(3.) Analyze the graph of the function based on increasing, decreasing, and constant intervals.
In other words: Based on the graph, at what intervals of x does the function, y increase. decrease, or remain constant?

$y \uparrow \;\;\;for\;\;\; x \in (-7, -1) \cup (1, 3) \cup (6, 8) \\[3ex] y \downarrow \;\;\;for\;\;\; x \in (-9, -7) \cup (-1, 1) \cup (3, 6)$
(4.) Complete the sentences below.
(a.) A set of points in the​ xy-plane is the graph of a function if and only if every​ _______ line intersects the graph in at most one point.

(b.) If the point ​(4,−​1) is a point on the graph of​ f, then​ f( __ ​)​ = ____.

(c.) If a function is defined by an equation in x and​ y, then the set of points​ (x,y) in the​ xy-plane that satisfies the equation is called the _______

(d.) The graph of a function $y = f(x)$ can have more than one of which type of​ intercept?

(a.) A set of points in the​ xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point.
This is known as the Vertical Line Test
A vertical line shows if two different​ y-values correspond to the same value of x for a set of points in the​ xy-plane.
If the vertical line intersects the graph at only one point, the graph is a function because no input, x has more than one output, y.
If the vertical line intersects the graph at more than point, the graph is not a function because there is at least an input, x that has more than one output, y.
Hence, the Vertical Line Test states that if a vertical line is drawn through the graph of a set of points in the rectangular coordinate system, the graph is a function if and only if the vertical line intersects the graph at only one point.

(b.) If the point ​(4,−​1) is a point on the graph of​ f, then​ f(4​)​ = −1

$For\;\;the\;\;point:\;\;(x, y) \\[3ex] y = f(x) \\[3ex] f(x) = y \\[3ex]$ (c.) If a function is defined by an equation in x and​ y, then the set of points​ (x,y) in the​ xy-plane that satisfies the equation is called the graph of the function.

(d.) Because of the Vertical Line Test: the graph of a function may intersect the y-axis only one time.
This implies that the graph of a function may not have more than one y-intercept.
Because two different input (two different x-values) can have the same output (same y-value), the graph of a function may have more than one x-intercept.

(5.)

(6.) Determine the value of p such that the point ​(−1​, 3​) is on the graph of ​$f(x) = px^2 + 5$.

$f(x) = px^2 + 5 \\[3ex] y = px^2 + 5 \\[3ex] For\;\;the\;\;point\;\;(-1, 3) \\[3ex] x = -1 \\[3ex] y = 3 \\[3ex] \implies \\[3ex] 3 = p(-1)^2 + 5 \\[3ex] 3 = p(1) + 5 \\[3ex] 3 = p + 5 \\[3ex] p + 5 = 3 \\[3ex] p = 3 - 5 \\[3ex] p = -2 \\[3ex]$
(7.) Use the graph to answer the questions.

$(a.)\;\; f(-14) \\[3ex] (b.)\;\; f(-4) \\[3ex] (c.)\;\; f(12) \\[3ex] (d.)\;\; f(0) \\[3ex] (e.)\;\; f(4) \\[3ex]$ (f.) For what value(s) of x is $f(x) = 0$?
Use a comma to separate answers as needed.

(g.) For what value(s) of x is $f(x) \gt 0$?
Type a compound inequality. Use a comma to separate answers as​ needed.

(h.) Write the domain of f in set notation.

(i.) Write the range of f in set notation.

(j.) What are the x-values of the​ x-intercept(s)?
Type an integer or a simplified fraction. Use a comma to separate answers as​ needed.

(k.) What are the y-values of the​ y-intercept(s)?
Type an integer or a simplified fraction. Use a comma to separate answers as​ needed.

(l.) How often does the line y = 1 intersect the​ graph?

(m.) How often does the line x = 5 intersect the​ graph?

(n.) For what​ value(s) of x does $f(x) = -6$?
Use a comma to separate answers as​ needed.

(o.) For what​ value(s) of x does $f(x) = 9$?
Use a comma to separate answers as​ needed.

$(a.)\;\; f(-14) = -6 \\[3ex] (b.)\;\; f(-4) = 6 \\[3ex] (c.)\;\; f(12 = 6) \\[3ex] (d.)\;\; f(0) = -3 \\[3ex] (e.)\;\; f(4) = -6 \\[3ex] (f.) \\[3ex] (-12, 0) \\[3ex] (-2, 0) \\[3ex] (8, 0) \\[3ex] \implies \\[3ex] f(x) = 0\;\;when\;\;x = -12, -2, 8 \\[3ex] (g.) \\[3ex] \underline{Excluded} \\[3ex] (-12, 0) \\[3ex] (-2, 0) \\[3ex] (8, 0) \\[3ex] \underline{Included} \\[3ex] (-6, 9) \\[3ex] (-4, 6) \\[3ex] (12, 6) \\[3ex] \implies \\[3ex] f(x) \gt 0\;\;for \\[3ex] -12 \lt x \lt -2, \;\;8 \lt x \le 12 \\[3ex] (h.) \\[3ex] Minimum\;\;x-value = -14 \\[3ex] Maximum\;\; x-value = 12 \\[3ex] D = \{x | -14 \le x \le 12\} \\[3ex] (i.) \\[3ex] Minimum\;\;y-value = -6 \\[3ex] Maximum\;\; y-value = 9 \\[3ex] R = \{y | -6 \le y \le 9\} \\[3ex] (j.) \\[3ex] x-intercepts = (-12, 0),\;\;(-2, 0),\;\; (8, 0) \\[3ex] x-values\;\;of\;\;the\;\;x-intercept = -12, -2, 8 \\[3ex] (k.) \\[3ex] y-intercept = (0, -3) \\[3ex] y-value\;\;of\;\;the\;\;y-intercept = -3 \\[3ex]$ (l.) Let us draw the line: $y = 1$ through the graph and count the number of times it intersects the graph.

There are three times (three vertical red lines) where the horizontal line $y = 1$ intersects the graph.

(m.) Let us draw the line: $x = 5$ through the graph and count the number of times it intersects the graph.

There is only one time (one horizontal green line) where the vertical line $x = 5$ intersects the graph.

$(n.) \\[3ex] (-14, -6) \\[3ex] (4, -6) \\[3ex] \implies \\[3ex] f(x) = -6\;\;when\;\;x = -14, 4 \\[3ex] (o.) \\[3ex] (-6, 9) \\[3ex] \implies \\[3ex] f(x) = 9\;\;when\;\;x = -6$
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