PRACTICAL
SECTION 1.7
FUNCTIONS AND GRAPHS
INTERMEDIATE ALGEBRA
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Date:
INTRODUCTION
In this lesson, we learn about functions. A function is represented by a formula. Into this, we substitute values
so that we can perform a computation. The only rule is that the computation can have only one result.
For example, the formula
𝑦 = √𝑥 + 4,
is a function.
The functions we study in algebra have two variables. The independent variable is usually x. We make
substitutions into x, and we are free to substitute any value (that doesn’t cause an error in the computation).
The dependent variable, y, is the result of the computation, and we call it dependent because its value depends
on x.
THE DOMAIN OF A FUNCTION
1. The domain of a function is simply the collection of values that we can substitute for x.
a. Negative numbers do not have (real) square roots. Think about the function, 𝑦 = √𝑥 + 4. Use
your knowledge about square roots, or your calculator, to decide the domain of this function.
Remember, you want to describe the values that can be substituted for x.
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b. Division by zero is not defined in mathematics. Think about the function, 𝑦 = 𝑥+3. Use what
you know about division to decide the domain of this function. Again, describe the values that
can be substituted for x.
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c. Cube roots are different from square roots. What is the domain for the function 𝑦 = √2 − 𝑥 ?
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PRACTICAL
SECTION 1.7
FUNCTIONS AND GRAPHS
INTERMEDIATE ALGEBRA
CHECK IT OUT!
For now, the only restrictions we need to worry about when finding the domain of a function are: we can’t
divide by zero, and we can’t take a square (or any even indexed) root of a negative number. This leads to the
following rules.



To find the domain of a function that has a variable denominator, set the denominator not equal to
zero, and solve.
To find the domain of a function that has a square (or other even indexed) root, set the expression
inside the radical greater than or equal to zero, and solve.
For now, any function that you encounter that does not have one of these restrictions has a domain
equal to all real numbers, which we symbolize as R.
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To find the domain of the function, 𝑦 = √3 − 2𝑥, we set 3 – 2x ≥ 0 and solve.
3 − 2𝑥 ≥ 0
−3
−3
−2𝑥 ≥ −3
−2𝑥 −3
≤
−2
−2
3
𝑥≤
2
So the domain is all values, x, where x ≤ 3/2, or x ≤ 1.5.
To find the domain of 𝑦 =
2𝑥−1
,
3𝑥−6
we set 3x – 6 ≠ 0, and solve. Adding 6 gives 3x ≠ 6, and dividing gives x ≠ 2. The
domain is all values, x, where x ≠ 2. Note that the numerator has no restrictions, so it is ignored.
The function, 𝑦 = 𝑥 2 − 7𝑥 + 1, has no radicals or division, so there are no restrictions, and the domain is all real
numbers, R.
2. Find the domain of the following functions.
a. 𝑦 = 5𝑥 − 𝑥 3
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PRACTICAL
SECTION 1.7
FUNCTIONS AND GRAPHS
INTERMEDIATE ALGEBRA
𝑥+1
b. 𝑦 = 2𝑥+10
c. 𝑦 = √2𝑥 + 8
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d. 𝑦 = √𝑥 − 4
GRAPHING FUNCTIONS
When we graph functions, we create a graphical representation of the relationship between the variables.
Remember, x is called the independent variable because it can be (and is) any value in the function’s domain.
When we graph the function, we graph a collection of representative points until we see a pattern. Once the
pattern is clear, we connect the points with a smooth curve.
3. The first function we considered was 𝑦 = √𝑥 + 4.
a. State the domain for this function.
b. Use your calculator to compute y coordinates that correspond to the x values that are
represented in the graph below. Square roots should be represented approximately.
x
-5
-4
-3
-2
0
5
y
c. One of the values in the table above resulted in an error. Why did this occur?
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PRACTICAL
SECTION 1.7
FUNCTIONS AND GRAPHS
INTERMEDIATE ALGEBRA
d. Graph the points computed above, and connect them with a smooth curve.
y
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
-5
-4
-3
-2
-1
-0.5
1
2
3
-1.0
4
5
x
e. Does your graph have a left endpoint? Explain your answer.
f.
Does your graph have a right endpoint? Explain your answer.
g. The x-intercept is the point where the graph touches the x-axis. Give the coordinates of this
point.
h. The y-intercept is the place where the graph touches the y-axis. Give the coordinates of this
point.
4. Our next function is 𝑦 = −2𝑥 + 4.
a. What is the domain for this function?
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PRACTICAL
SECTION 1.7
FUNCTIONS AND GRAPHS
INTERMEDIATE ALGEBRA
b. Make a table of points for the function.
x
-1
0
2
3
y
c. Graph the function below.
y
10
8
6
4
2
-4
-3
-2
-1
1
-2
2
3
4
x
-4
d. What are the x- and y-intercepts for this graph?
e. What is the y-coordinate of the x-intercept?
f.
What is the x-coordinate of the y-intercept?
CHECK IT OUT!


To find the x-intercept of a graph, set y = 0 in its equation and solve for x.
To find the y-intercept of a graph, set x = 0 in its equation and solve for x.
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PRACTICAL
SECTION 1.7
FUNCTIONS AND GRAPHS
INTERMEDIATE ALGEBRA
5. What are the x- and y-intercepts on the graph of the function, 𝑦 = 4𝑥 − 10?
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6. Think about the function 𝑦 = 𝑥−2.
a. What is the domain for this function?
b. Can the graph cross the place that is not in this function’s domain?
c. Fill out the table of points below for this function.
x
-8
-2
0
1
1.9
2.1
3
4
6
12
y
d. Graph this function below.
10
y
8
6
4
2
-8
-6
-4
-2
2
-2
4
6
8
10
12
x
-4
-6
-8
-10
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PRACTICAL
SECTION 1.7
FUNCTIONS AND GRAPHS
INTERMEDIATE ALGEBRA
FUNCTION NOTATION AND SUBSTITUTIONS
When working with functions, sometimes we need to describe substitutions in a simple way. The symbol, f(x)
(or g(x), h(x), etc.), is sometimes used instead of y for this purpose, but they represent the same thing. But what
does the f(x) notation mean?
Suppose we are working with the function, 𝑦 = √𝑥 + 2. If we want to describe substitutions, we might say that
y = f(x), or just 𝑓(𝑥) = √𝑥 + 2. If we want to know the value of y when x = 9, we substitute this into the
equation. Using function notation, we would write
𝑦 = 𝑓(9) = √9 + 2 = 3 + 2 = 5.
When we write f(9), we are describing a substitution. We are saying that x = 9 should be substituted into the
equation for y.
7. Suppose g(x) = 3x + 7.
a. What is g(6)?
b. What is g(-2)?
c. Sometimes we substitute variables into other variables, and describe this with function
notation. What would g(w) look like?
d. What would g(a + b) look like? Just do the substitution – the result won’t simplify nicely.
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