When it comes to rational functions, there are two ways to find the range depending on if the denominator is a linear expression or if it is a quadratic expression.

Let’s see the steps for each case.

Steps to determining the range of a rational function (denominator is a linear expression):

1. Find the inverse of the function

- Set the function equal to y, and solve for x

2. Find the domain of the inverse

- Substitute
at the end

3. Write the range

- Refer to previous blog post on domains if you need help with this

Steps to determining the range of a rational function (denominator is a quadratic expression):

- The domain of the inverse is the range of the function when you substitute y or
for x

1. Multiply the denominator to both sides of the equation

2. Find the discriminant in terms of y

- Set

3. Set the discriminant greater than or equal to 0 and solve for y

- discriminant=
, given

- Make a table to summarize the results if needed

4. Write the range

- Show when the factors of the discriminant and the discriminant are positive, negative, and 0

Let’s see an example for when the denominator is a linear expression (click here):

- The range is the set of y for which the discriminant is equal to or great than 0

Find the range of

1. Find the inverse of the function

2. Find the domain of the inverse

Domain: or ,or

3. Write the range

Range: or ,or

Now let’s see an example when the denominator is a quadratic expression (click here):

Find the range of

1. Multiply the denominator to both sides of the equation

2. Find the discriminant in terms of y

discriminant=

3. Set the discriminant greater than or equal to 0 and solve for y

4y is 0 when:y=0 y+4 is 0 when:y=-4

4y is negative when:y<0 y+4 is negative when:y<-4

4y is positive when:y>0 y+4 is positive when:y>-4

0 | + | ||||

0 | + | + | + | ||

4. Write the range

, , ,

Range: or , or

We’ll see one more example because it is tricky (click here):

Find the range of

1. Multiply the denominator to both sides of the equation

2. Find the discriminant in terms of y

Discriminant=

3. Set the discriminant greater than or equal to zero and solve for y

or

Note: We did not have to make a table because this was a simpler way to solve for y

4. Write the range

Range: , or

As you can see, finding the range of a function is trickier, especially finding the range of a rational function. It might seem hard and a little scary, but the more practice you get with this, the better you will become. For more help or practice on this topic, visit Symbolab’s Practice.

Until next time,

Leah

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