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Functions

2.1 What is a Function?

A function is a relationship between two sets with the restriction that for every element in the first set (the
x’s) there is only one value in the second set (the y’s.)

Example 1: The circle x2 + y2 = 25 is not a function since choosing x = 4 gives two values of y, y = 3 and
y = -3.

Example 2: The line 3x + 2y = 6 is a function since picking a value for x results in only a single value for
y. For instance, picking x = 4 gives one answer, y = -3.

Functions can also be represented by tables of values or by graphs.

Values of Functions

The value of a function is found by replacing the variable by the indicated value.

Example 3: For the function , find f (9) and f(a2) .

Solution :

Domain

The domain of a function is the set of values that can be used for the independent variable, x. Typically,
the domain is found by first determining the values that are not in the domain.

Example 4: Find the domain of .

Solution:

Since square roots can only be used on non- negative values , x - 3 ≥0. Therefore, the domain is
x ≥3 or x ∈ [3,∞).
Example 5: Find the domain of .

Solution:

Since the denominator of a fraction cannot be zero, x + 2 ≠ 0. Therefore, the domain is x ≠ -2
or x ∈ (-∞,-2) [ (-2,∞).

Range

The range of a function is the set of values that can be used for the dependent variable, y. To find the range
of a function, the easiest ways are to examine the function for ”obvious” features or to look at the graph of
the function
.

Example 6: Find the range of the function .

Solution:


The square root function always produces a non-negative result. Therefore the range of
  is the set of y ∈ [0,∞).

Example 7: Find the range of the function .

Solution:

Since the square root function always produces a non-negative result,
produces a value of 4 - something, such as 4, 3, 2, . . . . Therefore, the range is the set of
y  ∈(-∞, 4].

2.2 Graphs of Functions

Piecewise Continuous Functions


Piecewise continuous functions are constructed from two or more separate parts, as in the following.

Absolute Value

The absolute value function f (x) = lxl is really a piecewise continuous function.

The Greatest Integer Function

The greatest integer function is defined as

[[x]] = greatest integer less than or equal to x

Example 8: Find [[5]], [[2.8]] and [[-2.3]].

Solution:

The greatest integer function always round down to the nearest integer so we have [[5]] = 5,
[[2.8]] = 2, and [[-2.3]] = -3.

Another name for [[x]] is the floor function or . The floor function can be found on many calculators.

Families of Functions

A family of functions is a set of functions that has a specific feature that changes. For example, the family
given by f (x) = x2 + c is the set of parabolas that open upward and have a vertex on the y-axis.

2.3 Rate of Change

The average rate of change of a function is simply the slope of the line segment connecting two points on
the function.

A function is said to be increasing if the average rate of change is positive and decreasing if the average
rate of change is negative.

A function can be both increasing and decreasing. For example, y = x2 is decreasing on x ∈ (-∞, 0) and
increasing on x ∈ (0,∞).

2.4 Transformations of Functions

A transformation of a function means to change one or more characteristics of a function while keeping
the same basic function behavior. Typically, transformations are considered geometrically. Starting with a
function y = f (x),

• Vertical shift: y = f (x) + k moves the graph of f vertically by k units.
• Horizontal shift: y = f (x - h) moves the graph of f horizontally by h units.
• Flip vertically: y = -f (x) flips the graph of f upside-down.
• Stretch(shrink) vertically: y = c f (x) changes the height of the graph of f by a factor of c .
• Stretch(shrink) horizontally: y = f (cx) changes the width of the graph of f by a factor of .

Even and Odd Functions

A function y = f (x) is said to be even if f (-x) = f (x). A function y = f (x) is said to be odd if
f (-x) = -f (x). Even functions are symmetric to the y axis and odd functions are symmetric to the origin.

2.5 Quadratic Functions

A quadratic function has the form f (x) = ax2 + bx + c and has a parabola as its graph. Using the method
of completing the square, it can be converted to standard form

f (x) = a(x - h)2 + k, where (h, k) is the vertex of the parabola.

Example 9: Write the function f (x) = 3x2 - 7x + 2 in the form f (x) = a(x - h)2 + k.

Solution:


Using the method of completing the square from Chapter 1,

Extrema of a Quadratic

An extrema is a maximum or minimum point on a graph. The vertex of the quadratic f (x) = a(x - h)2 + k
is a minimum point if a > 0 and a maximum point if a < 0. A shortcut for finding the x- coordinate of the
vertex
is to use the vertex formula

Example 10: Find the vertex of f (x) = -0.02x2 + 7.1x + 14.3 and determine if the vertex is a maximum or
a minimum.

Solution:
Since the coefficient of x 2 is negative, we know that the parabola opens downward and
therefore, the vertex is a maximum point. The x-coordinate of the vertex is

Substituting back into the function, the y value is

f (177.5) = -0.02(177.5)2 + 7.1(177.5) + 14.3 = 644.425

Therefore, the vertex of the parabola is at (177.5, 644.425) and the vertex is a maximum point.
Extrema of Other Functions

To find the extrema of others functions, the simplest method is to look at the graph of the function.
However, this can be misleading.

Example 11: Use a calculator to locate the extrema of
f (x) = 0.1x5 + 0.53x4 - 0.438x3 - 4.832x2 - 0.376x + 11.904. You can assume that all the extrema are in
the interval -5≤ x ≤5.

Solution:

Start by sketching the graph in the window -5 ≤x ≤5, -10 ≤y ≤15. As seen in Figure 2.1,
there is something happening near x = -3 but it is not obvious that there is an extreme point.

Figure 2.1: Graph of f (x) using -5 ≤ x ≤ 5, -10 ≤ y ≤15

As seen in Figure 2.2, zooming in on the point eventually shows both a maximum and a
minimum point but it takes some effort.

Figure 2.2: Graph of f (x) using -3.3 ≤ x≤ -2.7, -0.002 ≤ y ≤ 0.002

 

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