Introduction to Functions - Math
Card 0 of 68
What is the range of 
?
What is the range of
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The range of a function is defined as the possible values for 
, or the possible outcomes. In this function, it is not possible to get any sort of negative number as an outcome. It is possible to get zero when 
. Thus, the range is all real numbers greater than or equal to 0.
The range of a function is defined as the possible values for
is a sine curve. What are the domain and range of this function?
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The domain includes the values that go into a function (the x-values) and the range are the values that come out (the 
or y-values). A sine curve represent a wave the repeats at a regular frequency. Based upon this graph, the maximum 
is equal to 1, while the minimum is equal to –1. The x-values span all real numbers, as there is no limit to the input fo a sine function. The domain of the function is all real numbers and the range is 
.
The domain includes the values that go into a function (the x-values) and the range are the values that come out (the
If 
, which of these values of 
is NOT in the domain of this equation?
If
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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:
Using
Therefore
What is the domain of the function?
What is the domain of the function?
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The domain of a function is the set of possible values for the 
variable. The range would be the possible values for the solution, 
.
The value inside of a square root must be greater than or equal to zero in order to have a real solution.
Now we can solve for 
in the inequality.
Square roots have positive and negative roots, so we need to set up two results. Remember to switch the direction fo the inequality for the negative solution.
These solutions can be combined to give our final answer.
Any values of 
that are not included will result in an imaginary (impossible) answer.
The domain of a function is the set of possible values for the
The value inside of a square root must be greater than or equal to zero in order to have a real solution.
Now we can solve for
Square roots have positive and negative roots, so we need to set up two results. Remember to switch the direction fo the inequality for the negative solution.
These solutions can be combined to give our final answer.
Any values of
Which of the following does NOT belong to the domain of the function $f(x)=\frac{x^{-1}$$$+$\sqrt{1-x}$}{4x^2$+1} ?
Which of the following does NOT belong to the domain of the function $f(x)=\frac{x^{-1}$$$+$\sqrt{1-x}$}{4x^2$+1} ?
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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 $0^{-1}$, 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.
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 $0^{-1}$, 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.
What is the domain of the function:
What is the domain of the function:
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The domain of a function consists of all of the possible values for x. In this case, we want to make sure that we are not dividing by 
(
in the denominator), since that would make our function undefined. Having 
would make the denominator 
. Thus, that is not in our domain. There is nothing else that would make this function undefined, and thus the domain is all real numbers except 
The domain of a function consists of all of the possible values for x. In this case, we want to make sure that we are not dividing by
Find the domain of the following function.
Find the domain of the following function.
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Domain represents all possible values for 
. In this situation, we want to make sure that the numbers under the square root are greater than or equal to 0. This will be the case when 
. Thus, the domain includes all real numbers such that 
.
Domain represents all possible values for
What is the range of 
?
What is the range of
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The range of a function is all of the possible 
values that the equation can take. For this equation our 
value cannot be negative, as a negative number squared still gives us a positive value.
Since 
, we know that the lowest possible value that 
can reach is 
. Therefore the range is 
.
The range of a function is all of the possible
Since
Which analysis can be performed to determine if an equation is a function?
Which analysis can be performed to determine if an equation is a function?
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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: 
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
The horizontal line test can be used to determine if a function is one-to-one, that is, if only one
Example of a function:
Example of an equation that is not a function:
Evaluate 
if 
and 
.
Evaluate
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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.
This expression is the same as saying "take the answer of
First, we need to find
Now we take this answer and plug it into
We can find the value of
This is our final answer.
Let 
and 
. What is 
?
Let
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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 
.
THe notation
The original expression for f(x) is
We will now distribute the -2 to the 2x - 1.
We must FOIL the
Now we collect like terms. Combine the terms with just an x.
Combine constants.
The answer is
If 
and 
, what is 
?
If
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means 
gets plugged into 
.
Thus 
.
Thus
Let 
and 
. What is 
?
Let
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Calculate 
and plug it into 
.
Calculate
Let 
. What is 
?
Let
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We are asked to find 
, which is the inverse of a function.
In order to find the inverse, the first thing we want to do is replace f(x) with y. (This usually makes it easier to separate x from its function.).
Next, we will swap x and y.
Then, we will solve for y. The expression that we determine will be equal to 
.
Subtract 5 from both sides.
Multiply both sides by -1.
We need to raise both sides of the equation to the 1/3 power in order to remove the exponent on the right side.
We will apply the general property of exponents which states that 
.
Laslty, we will subtract one from both sides.
The expression equal to y is equal to the inverse of the original function f(x). Thus, we can replace y with 
.
The answer is 
.
We are asked to find
In order to find the inverse, the first thing we want to do is replace f(x) with y. (This usually makes it easier to separate x from its function.).
Next, we will swap x and y.
Then, we will solve for y. The expression that we determine will be equal to
Subtract 5 from both sides.
Multiply both sides by -1.
We need to raise both sides of the equation to the 1/3 power in order to remove the exponent on the right side.
We will apply the general property of exponents which states that
Laslty, we will subtract one from both sides.
The expression equal to y is equal to the inverse of the original function f(x). Thus, we can replace y with
The answer is
What is the inverse of 
?
What is the inverse of
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The inverse of 
requires us to interchange 
and 
and then solve for 
.
Then solve for 
:
The inverse of
Then solve for
If 
, what is 
?
If
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To find the inverse of a function, exchange the 
and 
variables and then solve for 
.
To find the inverse of a function, exchange the
If the function 
is depicted here, which answer choice graphs 
?
If the function
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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 
.
The function
To translate along the y-axis, we use the function
What is the range of ?
What is the range of
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The range of a function is defined as the possible values for , or the possible outcomes. In this function, it is not possible to get any sort of negative number as an outcome. It is possible to get zero when . Thus, the range is all real numbers greater than or equal to 0.
The range of a function is defined as the possible values for
is a sine curve. What are the domain and range of this function?
Tap to see back →
The domain includes the values that go into a function (the x-values) and the range are the values that come out (the or y-values). A sine curve represent a wave the repeats at a regular frequency. Based upon this graph, the maximum is equal to 1, while the minimum is equal to –1. The x-values span all real numbers, as there is no limit to the input fo a sine function. The domain of the function is all real numbers and the range is .
The domain includes the values that go into a function (the x-values) and the range are the values that come out (the
If , which of these values of is NOT in the domain of this equation?
If
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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:
Using
Therefore