Functions - Pre-Calculus
Card 1 of 644
Which of the following is not true about a field. (Note: the real numbers 
is a field)
Which of the following is not true about a field. (Note: the real numbers
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It is not the case that for any element 
in a field, there is another one 
such that their product is 
. Take 
in the real numbers. Multiply 
by any number and you get 
, so you will never get 
. This is true for any field that has more than 1 element.
It is not the case that for any element
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What is the inverse function of
?
What is the inverse function of
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To find the inverse function of
we replace the 
with 
and vice versa.
So
Now solve for 
To find the inverse function of
we replace the
So
Now solve for
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What is the inverse of 
?
What is the inverse of
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When trying to find the inverse of a point, switch the x and y values.
So, 
When trying to find the inverse of a point, switch the x and y values.
So,
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Find the inverse of the following function:
Find the inverse of the following function:
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In order to find the inverse of the function, we need to switch the x- and y-variables.
After switching the variables, we have the following:
Now solve for the y-variable. Start by subtracting 10 from both sides of the equation.
Divide both sides of the equation by 4.
Rearrange and solve.
In order to find the inverse of the function, we need to switch the x- and y-variables.
After switching the variables, we have the following:
Now solve for the y-variable. Start by subtracting 10 from both sides of the equation.
Divide both sides of the equation by 4.
Rearrange and solve.
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Find the inverse of the function.
Find the inverse of the function.
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To find the inverse function, first replace 
with 
:
Now replace each 
with an 
and each 
with a 
:
Solve the above equation for 
:
Replace 
with 
. This is the inverse function:
To find the inverse function, first replace
Now replace each
Solve the above equation for
Replace
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Find the inverse of the function.
Find the inverse of the function.
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To find the inverse function, first replace 
with 
:
Now replace each 
with an 
and each 
with a 
:
Solve the above equation for 
:
Replace 
with 
. This is the inverse function:
To find the inverse function, first replace
Now replace each
Solve the above equation for
Replace
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Find the inverse of the function 
.
Find the inverse of the function
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To find the inverse of 
, interchange the 
and 
terms and solve for 
.
To find the inverse of
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What point is the inverse of the 
?
What point is the inverse of the
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When trying to find the inverse of a point, switch the x and y values.
So 
When trying to find the inverse of a point, switch the x and y values.
So
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Let
What does 
equal when 
?
Let
What does
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Because 3>0 we plug the x value into the bottom equation.
Because 3>0 we plug the x value into the bottom equation.
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Let
What does 
equal when 
?
Let
What does
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Because 
we use the first equation.
Therefore, plugging in x=0 into the above equation we get the following,
.
Because
Therefore, plugging in x=0 into the above equation we get the following,
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Determine the value of 
if the function is
Determine the value of
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In order to determine the value of 
of the function we set 
The value comes from the function in the first row of the piecewise function, and as such
In order to determine the value of
The value comes from the function in the first row of the piecewise function, and as such
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Determine the value of if the function is
Determine the value of
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In order to determine the value of of the function we set
The value comes from the function in the first row of the piecewise function, and as such
In order to determine the value of
The value comes from the function in the first row of the piecewise function, and as such
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For the function 
defined below, what is the value of 
when 
?
For the function 
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Evaluate the function for 
. Based on the domains of the three given expressions, you would use 
, since 
is greater than or equal to 
.
Evaluate the function for 
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Find the range of the following function:
Find the range of the following function:
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Every element of the domain has as image 7.This means that the function is constant . Therefore,
the range of f is :{7}.
Every element of the domain has as image 7.This means that the function is constant . Therefore,
the range of f is :{7}.
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What is the range of 
:
What is the range of
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We know that 
. So 
.
Therefore:
.
This gives:
.
Therefore the range is:
We know that
Therefore:
This gives:
Therefore the range is:
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Find the range of f(x) given below:
Find the range of f(x) given below:
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Note that: we can write f(x) as :
.
Since,
Therefore,
So the range is 
Note that: we can write f(x) as :
Since,
Therefore,
So the range is
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What is the range of 
:
What is the range of
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We have 
.
Adding 7 to both sides we have:
.
Therefore 
.
This means that the range of f is 
We have
Adding 7 to both sides we have:
Therefore
This means that the range of f is
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Find the domain of the following function:
Find the domain of the following function:
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The part inside the square root must be positive. This means that we must be 
. Thus
. Adding -121 to both sides gives 
. Finally multiplying both sides by (-1) give:
with x reals. This gives the answer.
Note: When we divide by a negative we need to flip our sign.
The part inside the square root must be positive. This means that we must be
Note: When we divide by a negative we need to flip our sign.
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What is the domain of the function given by:
What is the domain of the function given by:
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cos(x) is definded for all reals. cos(x) is always between -1 and 1. Thus 
. The value inside the square is always positive. Therefore the domain is the set of all real numbers.
cos(x) is definded for all reals. cos(x) is always between -1 and 1. Thus
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Find the domain of
Find the domain of
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Since 
for all real numbers, the denominator is never 0 .Therefore the domain is the set
of all real numbers.
Since
for all real numbers, the denominator is never 0 .Therefore the domain is the set
of all real numbers.
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