Representation of Functions
General Mathematics
Lesson Objectives
At the end of the lesson, the students must be
able to:
• define functions and related terms;
• determine if the given relation represents a
function;
• define piece-wise function; and
• represents real-life situations using functions,
including piece-wise functions.
Relation
A relation is a set of ordered pairs. The domain
of a relation is the set of first coordinates. The
range is the set of second coordinates.
Example of Relations
1. {(1, 4), (2, 5), (3, 6), (4, 8)}
2. {(4, 2), (4, -2), (9, 3), (9,3)}
3. {(1, a), (1, b), (1, c), (1,d)}
Functions
A function is a relation in which each element of
the domain corresponds to exactly one element
of the range.
Examples of Functions
1. {(1, 4), (2, 5), (3, 6), (4, 8)}
2. {(2, 1), (3, 1), (4, 1), (5,1)}
Example 1
Determine if the following relations represent a
function.
1. {(q, 0), (w, 1), (e, 2), (t, 3)}
2. {(-1, -2), (0, -2), (1, -2), (2, -2)}
3. {(1, 0), (1,1), (1, 2), (1, -2)}
4. {(x, 3), (y, 4), (z, 3), (w, 4)}
Some Types of Functions
Linear Function
A function f is a linear function if f(x) = mx + b,
where m and b are real numbers, and m and f(x) are
not both equal to zero.
Quadratic Function
A quadratic function is any equation of the form
f(x) = ax2+ bx + c where a, b, and c are real numbers
and a ≠ 0.
Some Types of Functions
Constant Function
A linear function f is a constant function if
f(x) = mx + b, where m = 0 and b is any real
number. Thus, f(x) = b.
Identity Function
A linear function f is an identity function if f(x) =
mx + b, where m = 1 and b = 0. Thus, f(x) = x.
Some Types of Functions
Absolute Value Function
The function f is an absolute value function if for all
real numbers x,
f(x) = x, for x ≥ 0
–x, for x ≤ 0
Piecewise Function
A piecewise function or a compound function is a
function defined by multiple sub-functions, where
each sub-function applies to a certain interval of
the main function's domain.
Example 2
To sell more T-shirts, the class needs to charge a
lower price as indicated in the following table:
The price for which you can sell x printed T-shirts is
called the price function p(x). p(x) represents each
data point in the table.
Solution to Example 2
Step 1:
Find the slope m of the line using the slope formula m = y2 – y1 / x2 – x1
Step 2:
Write the linear equation with two variables by substituting the values
of m and (x1, y1) to the formula y – y1 = m(x – x1)—the point-slope form
of a linear equation.
y – y1 = m(x – x1)
y – 540 = −15 (x − 500)
y – 540 = − 15 x + 100
y = − 15 x + 640
y = 640 – 0.2x
Thus, the price function is p(x) = 640 – 0.2x.
Example 3
Find the dimensions of the largest rectangular
garden that can be enclosed by 60 m of fencing.
Solution to Example 3
Let x and y denote the lengths of the sides of the garden.
Then the area A = xy must be given its maximum value.
Express A in terms of a single variable, either x or y. The
total perimeter is 60 meters.
2x + 2y = 60
x + y = 30
y = 30 – x
Hence,
A = xy
A = x(30 – x)
A= 30x – x2
Solution to Example 3
Write this equation in the vertex form by completing the
square.
A = –(x2 – 30x + 225) + 225
A = –(x – 15)2 + 225
The maximum area is 225 square meters.
Since x = 15 (the width) and 30 – x = 15 (the length), the
dimension that gives the maximum area is 15 meters by
15 meters.
Example 4
Sketch the graph of the given piecewise
function. What is f(– 4)? What is f(2)?
f(x) =
x + 2, if x ≥ 0
–x2+ 2, if x < 0
Solution to Example 4
To the right of the y-axis, the graph is a line
that has a slope of 1 and y-intercept of 2. To
the left of the y-axis, the graph of the function
is a parabola that opens downward and whose
vertex is (0, 2).
To sketch the graph of the
function, you can lightly draw
both graphs. Then darken the
portion of the graph that
represents the function.
Solution to Example 4
To find the value of the function when x = – 4,
use the second equation.
f(– 4) = – (– 4)2+ 2 = – 16 + 2 = – 14
To find the value of the function when x = 2,
use the first equation.
f(2) = 2 + 2 = 4
Exercise A
Determine whether or not each relation is a
function. Give the domain and range of each
relation.
1. {(2, 3), (4, 5), (6, 6)}
2. {(5, 1), (5, 2), (5, 3)}
3. {(6, 7), (6, 8), (7, 7), (7, 8)}
Exercise B
Tell whether the function described in each of
the following is a linear function, a constant
function, an identity function, an absolute value
function, ora piecewise function.
1. f(x) = 3x − 7
2. g(x) = 12
3. f(x) = 3, if x > −5
-6, if x < −5
Exercise B
Tell whether the function described in each of
the following is a linear function, a constant
function, an identity function, an absolute value
function, or a piecewise function.
4.
5.
Exercise C
A zumba instructor charges according to the number of
participants. If there are 15 participants or below, the
instructor charges ₱500.00 for each participant per
month. If the number of participants is between 15 and
30, he charges ₱400.00 for each participant per month. If
there are 30 participants or more, he charges ₱350.00 for
each participant per month.
1. Write the piecewise function that describes what the
instructor charges.
2. Graph the function.