Rational Functions

0:00 / 0:00

Rational Functions
A rational function can be expressed as:
f(x)=axn+bxn−1+...cxm+dxm−1+...\boxed{f(x)=\displaystyle\frac{ax^n+bx^{n-1}+...}{cx^m+dx^{m-1}+...}}f(x)=cxm+dxm−1+...axn+bxn−1+...​​
where nnn is the largest exponent in the numerator and mmm is the largest exponent in the denomiator. 

Properties of Rational Functions
X-Intercepts can be found by setting f(x)=0f(x)=0f(x)=0 and solving for xxx
Vertical asymptotes can be found by setting the denominator to 0 and solving for xxx
Horizontal asymptotes:
If n<m‾\underline{n<m}n<m​: There is a horizontal asymptote at y=0y=0y=0
If n=m‾\underline{n=m}n=m​: There is a horizontal asymptote at y=acy=\displaystyle\frac{a}{c}y=ca​
If n>m‾\underline{n>m}n>m​: There is no horizontal asymptote. There may be an oblique/diagonal asymptote.
If factoring the numerator & denominator of f(x)f(x)f(x), any terms that cancel out indicates where a missing point/hole in f(x)f(x)f(x)
The domain is the set of all real numbers except where there are vertical asymptotes & missing points/holes
The range is the set of all real numbers except where there are horizontal asymptotes in most cases
Watch Out!
Rational functions may only attain values strictly above or below horizontal asymptotes. This will affect the range.

PAGE BREAK
Example 1

Let's look at f(x)=2x−3x+1f(x)=\displaystyle\frac{2x-3}{x+1}f(x)=x+12x−3​.




xxx-Intercept: (32,0)\Bigg(\displaystyle\frac{3}{2},0\Bigg)(23​,0)
Vertical Asymptote: x=−1x=-1x=−1
Horizontal Asymptote: y=2y=2y=2
Missing Points/Holes: None
Intervals of Increasing: (−∞,−1) ∪ (−1,∞)(-\infin,-1)~\cup~(-1,\infin)(−∞,−1) ∪ (−1,∞)
Intervals of Decreasing: NA
Domain: (−∞,−1) ∪ (−1,∞)(-\infin,-1)~\cup~(-1,\infin)(−∞,−1) ∪ (−1,∞)
Range: (−∞,2) ∪ (2,∞)(-\infin,2)~\cup~(2,\infin)(−∞,2) ∪ (2,∞)

0:00 / 0:00

Solving Rational Equations
Rational equations are a type of equation containing at least one rational term, f(x)g(x)\boxed{\displaystyle\frac{f(x)}{g(x)}}g(x)f(x)​​, where f(x) & g(x)f(x)~\&~g(x)f(x) & g(x) are continuous polynomials. 

Rational equations can contain non-permissible values (NPV) that identify the restrictions of the domain. They identify the value of the variable that gives a 0 0~0 in the denominator. 

Wize Tip
The NPV's are equivalent to the vertical asymptote and missing points/holes.

PAGE BREAK
Example

Solve 2x−1=5x+2\displaystyle\frac{2}{x-1}=\displaystyle\frac{5}{x+2}x−12​=x+25​, stating any NPV's. 

Step 1. 
Identify any non-permissible values.

The NPV's are the vertical asymptotes. 
Therefore, x≠−2,1x\neq{-2,1}x=−2,1 otherwise there is a total of 000 in the denominator. 

Step 2. 
Find the lowest common denominator and multiply each term by it.  

The lowest common denominator is (x−1)(x+2)(x-1)(x+2)(x−1)(x+2). 

(2x−1=5x+2)×(x−1)(x+2)   →    2(x+2)=5(x−1)\Bigg(\displaystyle\frac{2}{x-1}=\displaystyle\frac{5}{x+2}\Bigg)\times(x-1)(x+2)~~~{\color{red}\rightarrow}~~~~2(x+2)=5(x-1)(x−12​=x+25​)×(x−1)(x+2)   →    2(x+2)=5(x−1)

Step 3. 
Solve for ′x′.\green{'x'.}′x′.

2(x+2)=5(x−1)2x+4=5x−53x=9∴ x=3\begin{array}{}
2(x+2)&=&5(x-1)\\\\
2x+4&=&5x-5\\\\
3x&=&9\\\\
\therefore~x&=&3

\end{array}2(x+2)2x+43x∴ x​====​5(x−1)5x−593​

Step 4. 
Verify.

2x−1=5x+223−1=53+222=551=1✓\begin{array}{}
\displaystyle\frac{2}{x-1}&=&\displaystyle\frac{5}{x+2}\\\\
\displaystyle\frac{2}{3-1}&=&\displaystyle\frac{5}{3+2}\\\\
\displaystyle\frac{2}{2}&=&\displaystyle\frac{5}{5}\\\\
1&=&1&\checkmark

\end{array}x−12​3−12​22​1​====​x+25​3+25​55​1​✓​

0:00 / 0:00

Example: Solving Rational Equations
Solve 3xx+5+1x−2=7x2+3x−10\displaystyle\frac{3x}{x+5}+\frac{1}{x-2}=\displaystyle\frac{7}{x^2+3x-10}x+53x​+x−21​=x2+3x−107​, stating any NPV's. 


Step 1. 
Identify any non-permissible values.

The NPV's are the vertical asymptotes. 

(x+5)=0→∴ x=−5(x−2)=0→∴ x=2x2+3x−10=(x+5)(x−2)=0→∴ x=−5,2\begin{array}{rccccccc}
(x+5)&=&0&&\rightarrow&&\therefore~x&=&-5\\\\
(x-2)&=&0&&\rightarrow&&\therefore~x&=&2\\\\
x^2+3x-10=(x+5)(x-2)&=&0&&\rightarrow&&\therefore~x&=&-5, 2

\end{array}(x+5)(x−2)x2+3x−10=(x+5)(x−2)​===​000​​→→→​​∴ x∴ x∴ x​===​−52−5,2​


Therefore, x≠−5,2x\neq{-5,2}x=−5,2 otherwise there is a total of 000 in the denominator. 

Step 2. 
Find the lowest common denominator and multiply each term by it.  

The lowest common denominator is (x+5)(x−2)(x+5)(x-2)(x+5)(x−2). 

(3xx+5+1x−2=7x2+3x−10)×(x+5)(x−2)      →      3x(x−2)+(x+5)=7\Bigg(\displaystyle\frac{3x}{x+5}+\frac{1}{x-2}=\displaystyle\frac{7}{x^2+3x-10}\Bigg)\times(x+5)(x-2)~~~~~~{\color{red}\rightarrow}~~~~~~3x(x-2)+(x+5)=7(x+53x​+x−21​=x2+3x−107​)×(x+5)(x−2)      →      3x(x−2)+(x+5)=7

Step 3. 
Solve for ′x′.\green{'x'.}′x′.

3x(x−2)+(x+5)=73x2−6x+x+5=73x2−5x−2=0(3x2−6x)+(x−2)=0(3x+1)(x−2)=0∴ x=−13,2\begin{array}{rcl}
3x(x-2)+(x+5)&=&7\\\\
3x^2-6x+x+5&=&7\\\\
3x^2-5x-2&=&0\\\\
(3x^2-6x)+(x-2)&=&0\\\\
(3x+1)(x-2)&=&0
\\\\\therefore~x&=&-\displaystyle\frac{1}{3}, 2

\end{array}3x(x−2)+(x+5)3x2−6x+x+53x2−5x−2(3x2−6x)+(x−2)(3x+1)(x−2)∴ x​======​77000−31​,2​

According to the NPV's, x≠2x\neq2x=2. 

Therefore, x=2x=2x=2 is an extraneous solution.

Step 4. 
Verify.

3xx+5+1x−2=7x2+3x−103(−13)(−13)+5+1(−13)−2=7(−13)2+3(−13)−10−314−37=−914−914=−914         ✓\begin{array}{rcl}
\displaystyle\frac{3x}{x+5}+\frac{1}{x-2}&=&\displaystyle\frac{7}{x^2+3x-10}\\\\
\displaystyle\frac{3(-\frac{1}{3})}{(-\frac{1}{3})+5}+\frac{1}{(-\frac{1}{3})-2}&=&\displaystyle\frac{7}{(-\frac{1}{3})^2+3(-\frac{1}{3})-10}
\\\\-\displaystyle\frac{3}{14}-\displaystyle\frac{3}{7}&=&-\displaystyle\frac{9}{14}\\\\
-\displaystyle\frac{9}{14}&=&-\displaystyle\frac{9}{14}~~~~~~~~~\checkmark

\end{array}x+53x​+x−21​(−31​)+53(−31​)​+(−31​)−21​−143​−73​−149​​====​x2+3x−107​(−31​)2+3(−31​)−107​−149​−149​         ✓​  


Therefore, x=−13x=-\displaystyle\frac{1}{3}x=−31​ is the only solution.

0:00 / 0:00

Example: Rational Functions
A graph of the function f(x)=x2−6x+8x2−3x−4f(x)=\displaystyle{\frac{x^2-6x+8}{x^2-3x-4}}f(x)=x2−3x−4x2−6x+8​ is sketched below:


Identify the equations for any vertical & horizontal asymptotes, and missing points/holes.

Factor f(x)f(x)f(x):

f(x)=x2−6x+8x2−3x−4f(x)=(x−4)(x−2)(x−4)(x+1)f(x)=x−2x+1\begin{array}{rcl}
f(x)&=&\displaystyle{\frac{x^2-6x+8}{x^2-3x-4}}\\\\
f(x)&=&\displaystyle{\frac{(x-4)(x-2)}{(x-4)(x+1)}}\\\\
f(x)&=&\displaystyle{\frac{x-2}{x+1}}

\end{array}f(x)f(x)f(x)​===​x2−3x−4x2−6x+8​(x−4)(x+1)(x−4)(x−2)​x+1x−2​​

Vertical Asymptotes: 

Let x+1=0x+1=0x+1=0. 

The vertical asymptote is at x=−1.x=-1.x=−1.

Horizontal Asymptote:

The degree of the numerator is equivalent to the degree in the denominator. 

Therefore, y=1y=1y=1.

Missing Points/Holes

Missing points/holes exist whenever terms in the numerator and denominator cancel. 

Since the term (x−4)(x-4)(x−4) was eliminated, there is a missing point/hole at x=4.x=4.x=4.

Given the equation 2x−3−34−x=2x−2x2−x−12\displaystyle\frac{2}{x-3}-\frac{3}{4-x}=\displaystyle\frac{2x-2}{x^2-x-12}x−32​−4−x3​=x2−x−122x−2​, answer the following questions.

Solve for xxx.

x=

(state the larger of the values)

x=

(state the smaller of the values)

I don't know

Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of: 

f(x)=x2−11x+18x2−3x+2f(x)=\displaystyle\frac{x^2-11x+18}{x^2-3x+2}f(x)=x2−3x+2x2−11x+18​

A) Vertical Asymptote

x=

B) Horizontal/Oblique Asymptote

y=

x

-Intercepts

x=

D) Missing Points (if there are none, put none)

x=

I don't know

Extra Practice

Rational Functions: Slant Asymptote

Find the slant asymptote to the curve y=x3+x2+1x2+1\displaystyle y=\frac{x^3+x^2+1}{x^2+1}y=x2+1x3+x2+1​. 

Q:\bf{Q:}Q: Find the exact list of vertical asymptotes of  f(x)=x−1x2+x\displaystyle f\left(x\right)=\frac{x-1}{x^2+x}f(x)=x2+xx−1​

Rational Functions: Slant Asymptote

Find the slant asymptote to the curve y=x3+x2+1x2+1\displaystyle y=\frac{x^3+x^2+1}{x^2+1}y=x2+1x3+x2+1​. 

 Q:\bf{Q:}Q: Find  f(g(x))f(g(x))f(g(x)) and its domain if f(x)=1x−1f\left(x\right)=\dfrac{1}{x-1}f(x)=x−11​ and g(x)=1x+2g\left(x\right)=\dfrac{1}{x+2}g(x)=x+21​

Solve the following inequality: 2x−1x−5≥x+1x+5\dfrac{2x-1}{x-5}\ge\dfrac{x+1}{x+5}x−52x−1​≥x+5x+1​

Rational Functions: Inequalities with Fractions

 Find all xxx that satisfy  x+3x2+8x+15≤0\dfrac{x+3}{x^2+8x+15}\le0x2+8x+15x+3​≤0

Rational Functions

 Find all xxx that satisfy xx−3≥4\dfrac{x}{x-3}\ge4x−3x​≥4

Rules of Exponents

 Simplify: −5x2ya33z5⋅37xby15z−2-\dfrac{5x^2y^a}{3^3z^5}\cdot\dfrac{3^7x^by}{15z^{-2}}−33z55x2ya​⋅15z−237xby​

Rational Functions

Let f(x)=x2+5x+6x−4f(x)=\dfrac{x^2+5x+6}{x-4}f(x)=x−4x2+5x+6​ be a rational function. 

Solving Rational Equations

Practice: Solving Rational Equations
Let f(x)=4x2−252x+5f(x)=\dfrac{4x^2-25}{2x+5}f(x)=2x+54x2−25​.  
If f(x)=4x2−2x+1f(x)=4x^2-2x+1f(x)=4x2−2x+1, then what is x?x?x?

Solving Rational Equations

Practice: Solving Rational Equations
Let f(x)=x2−4x−2f(x)=\dfrac{x^2-4}{x-2}f(x)=x−2x2−4​.  
If f(x)=x2−5x−6f(x)=x^2-5x-6f(x)=x2−5x−6, then what is x?x?x?

Solving Rational Equations

Practice: Solving Rational Equations
Given the equation 4xx−5+3x+2=xx2−3x−10\dfrac{4x}{x-5}+\dfrac{3}{x+2}=\dfrac{x}{x^2-3x-10}x−54x​+x+23​=x2−3x−10x​, answer the following questions.

Solving Rational Equations

Practice: Solving Rational Equations
Given the equation 4x+1−1x+3=x+2x2+4x+3\dfrac{4}{x+1}-\dfrac{1}{x+3}=\dfrac{x+2}{x^2+4x+3}x+14​−x+31​=x2+4x+3x+2​, answer the following questions.

Solving Rational Equations

Practice: Solving Rational Equations
True or false:
 8126\dfrac{81}{26}2681​ is a solution to 10=4x−13x+810=\dfrac{4x-1}{3x+8}10=3x+84x−1​.

Solving Rational Equations

Practice: Solving Rational Equations
True or false:
 177\dfrac{17}{7}717​ is a solution to −5=2x+3x−4-5=\dfrac{2x+3}{x-4}−5=x−42x+3​.

Solving Rational Equations

Solve for xxx: 
52x−1−73x+2=96x2+x−2\dfrac{5}{2x-1}-\dfrac{7}{3x+2}=\dfrac{9}{6x^2+x-2}2x−15​−3x+27​=6x2+x−29​

Solving Rational Equations

Solve 2x−3x2−3=5x+1\dfrac{2x-3}{x^2-3}=\dfrac{5}{x+1}x2−32x−3​=x+15​.

Graphing Rational Functions

Practice: Graphing Rational Functions
Write a rational function with the following properties:
An oblique asymptote at y=−3x+1y=-3x+1y=−3x+1

Graphing Rational Functions

Practice: Graphing Rational Functions
Write a rational function with the following properties:
An oblique asymptote at y=2x−7y=2x-7y=2x−7

Graphing Rational Functions

Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d}f(x)=cx+dax+b​ that describes the following:
Vertical asymptote at x=−1x=-1x=−1

Graphing Rational Functions

Practice: Graphing Rational Functions
Find a rational function of the form f(x)=ax+bcx+df(x)=\displaystyle\frac{ax+b}{cx+d}f(x)=cx+dax+b​ that describes the following:
Vertical asymptote at x=−2x=-2x=−2

Graphing Rational Functions

Practice: Graphing Rational Functions
Sketch a graph of f(x)=x2−25x+5f(x)=\dfrac{x^2-25}{x+5}f(x)=x+5x2−25​, finding all asymptotes, x-intercepts, missing points, and positive & negative intervals. 

Graphing Rational Functions

Practice: Graphing Rational Functions
Sketch a graph of f(x)=−x+9x+3f(x)=\dfrac{-x+9}{x+3}f(x)=x+3−x+9​, finding all asymptotes, intercepts, missing points, and positive & negative intervals. 

Graphing Rational Functions

Practice: Graphing Rational Functions
A rational function has the following properties:
A horizontal asymptote at y=0y=0y=0.

Graphing Rational Functions

Practice: Graphing Rational Functions
A rational function has the following properties:
A horizontal asymptote at y=0y=0y=0.

Rational Functions

Practice: Rational Functions
Match the equation for the rational function with its appropriate properties. 

Rational Functions

Practice: Rational Functions
Match the equation for the rational function with its appropriate properties. 

Rational Functions

Practice: Rational Functions
Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of: 
f(x)=3x2+14x+8x2−2x−14f(x)=\dfrac{3x^2+14x+8}{x^2-2x-14}f(x)=x2−2x−143x2+14x+8​

Rational Functions

Practice: Rational Functions
Identify the vertical asymptotes, the horizontal asymptotes, the x-intercepts, and any missing points of: 
f(x)=2x2+x−12x2+5x−3f(x)=\dfrac{2x^2+x-1}{2x^2+5x-3}f(x)=2x2+5x−32x2+x−1​

Rational Functions

Practice: Rational Functions
Match the rational function with the correct sketch of its graph. 

Rational Functions

Practice: Rational Functions
Match the rational function with the correct sketch of its graph. 

Rational Functions

Which of the following graphs is a good approximation of f(x)=x2−3xx+6f(x) = \frac{x^2 - 3x}{x + 6}f(x)=x+6x2−3x​ for xxx on the small interval [−1/10,1/10][-1/10, 1/10][−1/10,1/10]

Rational Functions

Match the following five functions with the above graphs.

Rules of Exponents

 Simplify: −5x2ya33z5⋅37xby15z−2-\dfrac{5x^2y^a}{3^3z^5}\cdot\dfrac{3^7x^by}{15z^{-2}}−33z55x2ya​⋅15z−237xby​

Rational Functions

Q:\bf{Q:}Q: Find the exact list of vertical asymptotes of  f(x)=x−1x2+x\displaystyle f\left(x\right)=\frac{x-1}{x^2+x}f(x)=x2+xx−1​

Rational Functions: Slant Asymptote

Find the slant asymptote to the curve y=x3+x2+1x2+1\displaystyle y=\frac{x^3+x^2+1}{x^2+1}y=x2+1x3+x2+1​. 

Rational Functions: Inequalities with Fractions

 Find all xxx that satisfy  x+3x2+8x+15≤0\dfrac{x+3}{x^2+8x+15}\le0x2+8x+15x+3​≤0

Rational Functions

 Find all xxx that satisfy xx−3≥4\dfrac{x}{x-3}\ge4x−3x​≥4

Solve the following inequality: 2x−1x−5≥x+1x+5\dfrac{2x-1}{x-5}\ge\dfrac{x+1}{x+5}x−52x−1​≥x+5x+1​

 Q:\bf{Q:}Q: Find  f(g(x))f(g(x))f(g(x)) and its domain if f(x)=1x−1f\left(x\right)=\dfrac{1}{x-1}f(x)=x−11​ and g(x)=1x+2g\left(x\right)=\dfrac{1}{x+2}g(x)=x+21​

Rational Functions

Solve the following inequality: 3x+1x−2≤2x−1x+2\frac{3x+1}{x-2}\leq \frac{2x-1}{x+2}x−23x+1​≤x+22x−1​

Rational Functions

Solve the following inequality for xxx
x−1x+1≤x+2x−1\frac{x-1}{x+1} \leq \frac{x+2}{x-1}x+1x−1​≤x−1x+2​

Factoring and Cancelling terms in a Rational Function

When x=2.1x = 2.1x=2.1 , the function
f(x)=3x2−3x−6x3−2x2−x+2f(x) = \frac{3x^2 - 3x  - 6}{x^3 - 2x^2 - x + 2}f(x)=x3−2x2−x+23x2−3x−6​
is closest to which number?

Evaluating Rational Functions at a Point

When x=−100x = -100x=−100 , the function
f(x)=−3x16−2x10−x3+215x13+2x10−x5+3f(x) = \frac{-3 x^{16} - 2 x^{10} - x^3 + 2}{15 x^{13} + 2 x^{10} - x^5 + 3}f(x)=15x13+2x10−x5+3−3x16−2x10−x3+2​
is closest to which number?

Rational Functions

When x=0.01x = 0.01x=0.01 the function 
f(x)=3x9+7x3−105x8−3x7+2xf(x) = \frac{3x^9 + 7x^3 - 10}{5x^8 -3x^7 + 2x}f(x)=5x8−3x7+2x3x9+7x3−10​
is closest to which number?

Rational Functions

When x=1000x = 1000x=1000 the function 
f(x)=3x9+7x3−105x8−3x7+2xf(x) = \frac{3x^9 + 7x^3 - 10}{5x^8 -3x^7 + 2x}f(x)=5x8−3x7+2x3x9+7x3−10​
is closest to which number?

Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Rational Functions

Popular Courses

Rational Functions

Properties of Rational Functions

Solving Rational Equations

Example: Solving Rational Equations

Example: Rational Functions

Extra Practice

Rational Functions: Slant Asymptote

Rational Functions

Rational Functions: Slant Asymptote

Rational Functions: Inequalities with Fractions

Rational Functions

Rules of Exponents

Rational Functions

Solving Rational Equations

Solving Rational Equations

Solving Rational Equations

Solving Rational Equations

Solving Rational Equations

Solving Rational Equations

Solving Rational Equations

Solving Rational Equations

Graphing Rational Functions

Graphing Rational Functions

Graphing Rational Functions

Graphing Rational Functions

Graphing Rational Functions

Graphing Rational Functions

Graphing Rational Functions

Graphing Rational Functions

Rational Functions

Rational Functions

Rational Functions

Rational Functions

Rational Functions

Rational Functions

Rational Functions

Rational Functions

Rules of Exponents

Rational Functions

Rational Functions: Slant Asymptote

Rational Functions: Inequalities with Fractions

Rational Functions

Rational Functions

Rational Functions

Factoring and Cancelling terms in a Rational Function

Evaluating Rational Functions at a Point

Rational Functions

Rational Functions