High School Solutions – Functions Calculator, Range (Part II)

Last blog post, we talk about how to find the range of linear, radical, and quadratic functions and what a range is. This week we will learn how to find the range of rational functions, which is trickier.

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 

• Substitute y = x at the end

2.  Find the domain of the inverse

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

3.  Write the range

• The domain of the inverse is the range of the function when you substitute y or \f(x) for x

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

1. Multiply the denominator to both sides of the equation

• Set \f(x)=y

2. Find the discriminant in terms of y

• discriminant= b^2-4ac, given ax^2+bx+c=0

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

• Make a table to summarize the results if needed 

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

4. Write the range

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

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

                                              Find the range of \y=\frac{x+3}{x-4}

1. Find the inverse of the function

                                                           \y=\frac{x+3}{x-4}

                                                        yx-4y-3=x

                                                      -4y-3=x(1-y)

                                                         x=\frac{-4y-3}{1-y}

                                                       y^{-1}=\frac{-4x-3}{1-x}

2. Find the domain of the inverse

                                                                1-x=0

                                                                  x=1

                                         Domain: x<1 or x>1,or (-∞,1)∪(1,∞)

3. Write the range

                                          Range: y<1 or y>1,or (-∞,1)∪(1,∞)

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

                                           Find the range of \f(x)=\frac{4}{x^2-2x}

1. Multiply the denominator to both sides of the equation

                                                   \y=\frac{4}{x^2-2x}

                                                          y(x^2-2x)=4

2. Find the discriminant in terms of y

                                                      yx^2-2yx-4=0

                               discriminant= (-2y)^2-4(y)(-4)=4y^2+16y

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

                                                      4y^2+16y≥0

                                                        4y(y+4)≥0

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

	y<-4	y=-4	-4<y<0	y=0	y>0
y	-	-	-	0	+
y+4	-	0	+	+	+
y(y+4)	-\:∙\:-\:=+	-\:∙\:0\:=0	-\:∙\:+\:=-	0\:∙\:+\:=0	+\:∙\:+\:=+

4. Write the range

                                               y<-4 , y=-4, y=0, y>0

                                   Range:y≤-4 or y≥0, or (-∞,-4)∪(0,∞)

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

                                               Find the range of y=\frac{x}{x^2+4}

1. Multiply the denominator to both sides of the equation

                                                            y(x^2+4)=x

2. Find the discriminant in terms of y

                                                           yx^2-x+4y=0

                                     Discriminant= (-1)^2-4(y)(4y)=1-16y^2

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

                                                            1-16y^2≥0

                                                              1≥16y^2

                                                              \frac{1}{16}≥y^2

                                                        y≥\frac{-1}{4}  or  y≤\frac{1}{4}

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

4. Write the range

                                              Range: \frac{-1}{4}≤y≤\frac{1}{4}, or [\frac{-1}{4},\frac{1}{4}]

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