Introduction to Functions
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Math › Introduction to Functions
Which analysis can be performed to determine if an equation is a function?
Vertical line test
Horizontal line test
Calculating zeroes
Calculating domain and range
Explanation
The vertical line test can be used to determine if an equation is a function. In order to be a function, there must only be one (or
) value for each value of
. The vertical line test determines how many
(or
) values are present for each value of
. If a single vertical line passes through the graph of an equation more than once, it is not a function. If it passes through exactly once or not at all, then the equation is a function.
The horizontal line test can be used to determine if a function is one-to-one, that is, if only one value exists for each
(or
) value. Calculating zeroes, domain, and range can be useful for graphing an equation, but they do not tell if it is a function.
Example of a function:
Example of an equation that is not a function:
Which analysis can be performed to determine if an equation is a function?
Vertical line test
Horizontal line test
Calculating zeroes
Calculating domain and range
Explanation
The vertical line test can be used to determine if an equation is a function. In order to be a function, there must only be one (or
) value for each value of
. The vertical line test determines how many
(or
) values are present for each value of
. If a single vertical line passes through the graph of an equation more than once, it is not a function. If it passes through exactly once or not at all, then the equation is a function.
The horizontal line test can be used to determine if a function is one-to-one, that is, if only one value exists for each
(or
) value. Calculating zeroes, domain, and range can be useful for graphing an equation, but they do not tell if it is a function.
Example of a function:
Example of an equation that is not a function:
Evaluate if
and
.
Undefined
Explanation
This expression is the same as saying "take the answer of and plug it into
."
First, we need to find . We do this by plugging
in for
in
.
Now we take this answer and plug it into .
We can find the value of by replacing
with
.
This is our final answer.
If the function is depicted here, which answer choice graphs
?
Explanation
The function shifts a function f(x)
units to the left. Conversely,
shifts a function f(x)
units to the right. In this question, we are translating the graph two units to the left.
To translate along the y-axis, we use the function or
.
Evaluate if
and
.
Undefined
Explanation
This expression is the same as saying "take the answer of and plug it into
."
First, we need to find . We do this by plugging
in for
in
.
Now we take this answer and plug it into .
We can find the value of by replacing
with
.
This is our final answer.
If the function is depicted here, which answer choice graphs
?
Explanation
The function shifts a function f(x)
units to the left. Conversely,
shifts a function f(x)
units to the right. In this question, we are translating the graph two units to the left.
To translate along the y-axis, we use the function or
.
Let and
. What is
?
Explanation
THe notation is a composite function, which means we put the inside function g(x) into the outside function f(x). Essentially, we look at the original expression for f(x) and replace each x with the value of g(x).
The original expression for f(x) is . We will take each x and substitute in the value of g(x), which is 2x-1.
We will now distribute the -2 to the 2x - 1.
We must FOIL the term, because
.
Now we collect like terms. Combine the terms with just an x.
Combine constants.
The answer is .
Let and
. What is
?
Explanation
THe notation is a composite function, which means we put the inside function g(x) into the outside function f(x). Essentially, we look at the original expression for f(x) and replace each x with the value of g(x).
The original expression for f(x) is . We will take each x and substitute in the value of g(x), which is 2x-1.
We will now distribute the -2 to the 2x - 1.
We must FOIL the term, because
.
Now we collect like terms. Combine the terms with just an x.
Combine constants.
The answer is .
Which of the following does NOT belong to the domain of the function ?
0
-1
-1/2
1/2
1
Explanation
The domain of a function includes all of the values of x for which f(x) is real and defined. In other words, if we put a value of x into the function, and we get a result that isn't real or is undefined, then that value won't be in the domain.
If we let x = 0, then we will be forced to evaluate , which is equal to 1/0. The value of 1/0 is not defined, because we can never have zero in a denominator. Thus , because f(0) isn't defined, 0 cannot be in the domain of f(x).
The answer is 0.
If , which of these values of
is NOT in the domain of this equation?
Explanation
Using as the input (
) value for this equation generates an output (
) value that contradicts the stated condition of
.
Therefore is not a valid value for
and not in the equation's domain: