Functions: Domain, Range and Variation |
From our discussion of composite functions in the previous lesson, we learned that the range of a function can become the domain of another function. Therefore, when working with composite functions, it is important to define the range of those functions.
For instance, in the question about shoe sizes, the output or range values for f (x) determine the input or domain values for g (x).
The smallest woman's shoe size in America is 4 and the largest is 12, therefore the range
for f (x), the function which converts American sizes to European sizes, is .
Similarly, g(x), the function which converts European sizes to British sizes, has
domain , and range .
We found h(x) = g [ f (x) ] by first applying f and then applying g.
g [ f (x) ] = g [ 2(x + 12) ] = g (2x + 24) = ½ (2x + 24) – 14 = x + 12 – 14 = x – 2.
It is obvious from the chart that the range of f (x) is the domain of g(x).
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Methods for defining the range of a function:
Example 1
State the domain and range of :
a) f (x) = – 2x + 3; with | b) g(x) = 3x + 12 |
c) f (x) = x² + 3x – 10 | d) |
e) | f) |
Solution
a) f (x) = – 2x + 3 is a linear function. The domain is given .
Since this line has a negative slope, the maximum range value of 6 occurs at x = – 1 and the minimum range value of – 7 occurs at x = 5.
Domain: | Range: |
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b) g(x) = 3x + 12 is also a linear function but since no domain interval has been specified, the domain is all real numbers (). Since this is true, the range is also all real numbers.
Domain: | Range: |
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c) f (x) = x²+ 3x – 10 is a quadratic function (its graph is a parabola) and since the coefficient of x² (1), is positive, the parabola will open upwards.
Therefore, f (x) will take on a minimum value at the vertex.
Since x² + 3x – 10 = (x + 5)(x – 2), the zeros occur at x = – 5 and x = 2. Therefore, the axis of symmetry is the line x = . When we substitute x = into x² + 3x – 10, we get .
Domain: | Range: . |
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d) is a first degree rational function. Since the denominator cannot = 0,
the domain is .
To find the range, we divide every term in the fraction by the highest power of the variable in the fraction, then set x = . Any fraction with an infinite denominator = 0.
so , and when we set x =, 4/x = 0, so we get ½.
Domain: | Range: |
Note: when we graph this function, there will be a vertical asymptote at x = 2 and a horizontal asymptote at y = ½. For more on asymptotes see the lesson on Rational Functions.
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e) is also a rational function, so we must eliminate the
values of x that make the denominator = 0 from the domain.
Thus, is the domain.
To find the range, we do as in (d) but this time we divide through by x², the highest power of x in the fraction.
We get , and when we set x = , the fraction = 0/1 = 0.
Domain: | Range: |
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f) is a function with an even root, therefore 5 – 5x² must be 0.
Solving , we get
5 – 5x² is a parabola opening down; maximum at x = 0,
axis of symmetry: the y-axis; range is .
Domain: | Range: |
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range of a function | periodic functions | increasing, decreasing, constant |
practice | solutions |
Certain mathematical functions repeat themselves across their domain. The y-values occur within a precise range over and over again. Such functions are called periodic functions.
Here are the graphs of two periodic functions.
We see that both graphs repeat their y-values every 4 units of the x-axis.
We say that the period of these functions is 4 units.
The range of f (x) on the left is , and
the range of g(x) on the right is .
A periodic function f has the property that the y-values are the same for x_{ 1} and x_{ 2} (in the domain). The interval x_{ 2} – x_{ 1} is called the period of function f. |
The trigonometric functions discussed in the Trig MathRoom are periodic functions.
range of a function | periodic functions | increasing, decreasing, constant |
practice | solutions |
Variation
Increasing, Decreasing and Constant Functions
On intervals where the graph of a function rises consistently from left to right, the function is increasing. If y gets bigger as x gets bigger, the function is increasing.
On intervals where the graph of a function falls consistently from left to right, the function is decreasing. If y gets smaller as x gets bigger, f (x) is decreasing.
A function which neither rises nor falls is a constant function.
If f(x) increases as x increases, f is an increasing function. If f(x) decreases as x increases, f is a decreasing function. If all values of f(x) are the same, f is a constant function. |
Example 2
Determine which functions are increasing and which are decreasing.
A linear function with a positive slope is increasing.
A linear function with a negative slope is decreasing.
A linear function parallel to the x-axis (slope = 0) is a constant function.
Some functions, such as the quadratic function (a parabola), are increasing on one part of their domain and decreasing on another part. Such functions are neither increasing nor decreasing, since they do not consistently rise or fall over their entire domain.
With such functions, we state the intervals where they rise and fall
with reference to the domain elements (x's).
For instance, y = x² – 2 is decreasing for all x < 0 and increasing for all x > 0,
since it is a parabola opening upwards with vertex on the y-axis.
Example 3
Determine the intervals for which these functions are increasing, decreasing and constant.
Solution
(a) decreasing for x < 0, increasing for x > 0
(b) increasing for x < – 3 and x > 0, decreasing for – 3 < x < 0
(c) decreasing for x < 0, increasing for x > 0
(d) increasing for x < 1, decreasing for x > 1
(e) increasing for all x ; at x = 2, this function is undefined.
(f) constant for x < – 2, decreasing for – 2 < x < 0, increasing for x > 0.
Note: The dotted lines in (e) at x = 2 and y = 1 are vertical and horizontal asymptotes.
Values of the domain and range approach x = 2 and y = 1, but never equal them.
range of a function | periodic functions | increasing, decreasing, constant |
practice | solutions |
1) State the domain and range of these functions:
a) f (x) = 2x – 5; | b) f (x) = x² – 2x + 1 | c) g(x) = | x + 3 | |
d) | e) g(x) = 3 – 2x – x² | f) h(x) = 5 |
2) Determine the period and the range for this function:
3) Determine the intervals for which these functions are increasing, decreasing and constant.
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range of a function | periodic functions | increasing, decreasing, constant |
practice | solutions |
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1) State the domain and range of these functions:
a) f (x) = 2x – 5; d: , r: |
b) f (x) = x² – 2x + 1 d: , r: |
c) g(x) = | x + 3 | d: , r: |
d) d: , r: |
e) g(x) = 3 – 2x – x² d: , r: |
f) h(x) = 5 d: , r: h(x) = 5 |
2) Determine the period and the range for this function:
3)
increasing | decreasing | constant |
a) x < 2 | x > 2 | |
b) | x < – 2 union with x > 1 | |
c) | x < – 3 union with x > 3 | |
d) x > – 1 | x < – 1 | |
e) for all x, | ||
f) | x > 0 | x < – 2 |
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range of a function | periodic functions | increasing, decreasing, constant |
practice | solutions |