Introduction to Functions - Algebra 2

The function includes all possible y-values (outputs). There is nothing you can put in for y that won't work.

A number taken to a power must be positive.

The square root of any number cannot be negative.

The absolute value of a number cannot be negative.

To create the inverse, switch x and y making the solution x=3y+3.

y must be isolated to finish the problem.

The domain is all the possible values of . To determine the domain, we need to determine what values don't work for this equation. The only value that is not allowed for this equation is 5, since that would make the denominator have a value of , and you can not divide by . Therefore, the domain of this equation is:

and

The domain of any quadratic function is always all real numbers.

The range of this function is anything greater than or equal to 5.

These written in the correct notation is:

Soft brackets are needed for infinity and a hard square bracket for 5 because it is included in the solution.

Because the domain is giving us a wide range of values, we can easily eliminate the fractional function as it only isolates a single value. We can eliminate as it means I am restricted to as my domain but I am looking for domain values greater than . This leaves us with the radical functions.

We have to remember the smallest possible value inside the radical is zero. Anything less means we will be dealing with imaginary numbers.

, This means the domain is which doesn't match our domain so this is wrong.

, . This means the domain is which doesn't match our domain since we want to EXCLUDE so this is wrong.

Since this is fractional expression with a radical in the denominator, we need to remember the bottom can't be zero and just set that denominator to equal . Square both sides to get . This actually means is not acceptable but any values greater than that is good. This is the correct answer.

Plug 3 in for x:

Simplify:

=

= 5

All inputs are valid. There is nothing you can put in for x that won't work.

All inputs are valid. There is nothing you can put in for x that won't work.

To find the domain, find all areas of the number line where the fraction is defined.

because the denominator of a fraction must be nonzero.

Factor by finding two numbers that sum to -2 and multiply to 1. These numbers are -1 and -1.

The domain is the set of x-values that make the function defined.

This function is defined everywhere except at , since division by zero is undefined.

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 .

All inputs are valid. There is nothing you can put in for x that won't work.

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:

In order for the function to be real, the value inside of the square root must be greater than or equal to zero. The domain refers to the possible values of the independent variable (x-value) that allow this to be true.

For this term to be real, must be greater than or equal to zero.

A function has to pass the vertical line test, which means that a vertical line can only cross the function one time. To put it another way, for any given value of , there can only be one value of . For the function , there is one value for two possible values. For instance, if , then . But if , as well. This function fails the vertical line test. The other functions listed are a line,, the top half of a right facing parabola, , a cubic equation, , and a semicircle, . These will all pass the vertical line test.

This function is a parabola that has been shifted up five units. The standard parabola has a range that goes from 0 (inclusive) to positive infinity. If the vertex has been moved up by 5, this means that its minimum has been shifted up by five. The first term is inclusive, which means you need a "\[" for the beginning.

Minimum: 5 inclusive, maximum: infinity

Range: