09/14 9. Rational Expressions and Functions

Expressions, restrictions, and simplification

A rational expression extends an ordinary fraction by allowing polynomials in the numerator and denominator. Its central condition is that the denominator cannot be zero:

To find restrictions, set the original denominator equal to zero, solve, and exclude those values from the domain. For example, in

factor the denominator:

Thus, $x\ne 3$ and $x\ne -3$, and the domain is

When simplifying, factor the numerator and denominator completely, record the original restrictions, cancel common factors, and write the reduced expression. For example,

but the restrictions from the original denominator remain: $x\ne 2,-3$. The canceled factor creates a missing point rather than restoring $x=2$ to the domain.

A crucial distinction is that factors may be canceled, but individual terms may not. In a product, factor first and then cancel:

with original restrictions $x\ne 1,3,-3$.

Takeaway: Simplification changes the formula, not the original domain restrictions.

Operations with rational expressions

Multiplication uses the product of the numerators and the product of the denominators, followed by factoring and cancellation:

Division means multiplication by the reciprocal:

Every factor that appears in a denominator, including a factor introduced by taking a reciprocal, must be nonzero.

Addition and subtraction require a common denominator. For example,

has least common denominator $(x-1)(x+2)$. Rewrite each fraction:

The restrictions are $x\ne 1,-2$. Do not add denominators; instead, create a common denominator and combine only the numerators.

Takeaway: Factor before multiplying or dividing, and use a common denominator before adding or subtracting.

Solving rational equations and proportions

A rational equation contains one or more rational expressions. Use a systematic process:

1. Find all restrictions from the original denominators.

2. Determine the least common denominator.

3. Multiply every term on both sides by the least common denominator.

4. Solve the resulting equation.

5. Check each candidate in the original equation.

6. Reject restricted values and any value that fails the original equation.

For example, solve

The restrictions are $x\ne 0,1$, and the least common denominator is $x(x-1)$. Multiplying every term by it gives

After simplifying,

The quadratic formula gives

Neither value is restricted, and both satisfy the original equation. Therefore,

Multiplying by an expression that might equal zero can produce a candidate that was not valid in the original equation, so checking is essential.

A proportion is a special rational equation. For

cross multiplication gives $8x=60$, so $x=7.5$.

Takeaway: Clearing denominators is efficient only when restrictions are recorded first and every result is checked afterward.

Features of rational functions

To analyze a rational function, first factor its numerator and denominator and reduce common factors while preserving all original restrictions. Then identify intercepts, holes, vertical asymptotes, and end behavior.

For

an $x$-intercept occurs where the reduced numerator is zero and the reduced denominator is nonzero. The $y$-intercept is $R(0)$, provided $0$ is in the domain.

Consider

The zeros of the numerator give $x$-intercepts at $(2,0)$ and $(-1,0)$. At $x=0$,

so the $y$-intercept is $(0,\frac{1}{6})$.

For a canceled factor, substitute its restricted input into the reduced formula to find the hole's coordinates. For example,

The factor $x-2$ creates a hole. Its output is

so the hole is $(2,-3)$. The remaining denominator factor creates the vertical asymptote $x=3$.

Takeaway: Canceled denominator factors identify holes; uncanceled denominator factors identify vertical asymptotes.

End behavior and asymptotes

After reducing common factors, compare the degrees of the numerator and denominator to determine horizontal or slant end behavior.

• If $\deg p<\deg q$, the horizontal asymptote is $y=0$.

• If $\deg p=\deg q$, the horizontal asymptote is the ratio of the leading coefficient of the numerator to the leading coefficient of the denominator.

• If $\deg p=\deg q+1$, polynomial division gives a slant asymptote.

• If $\deg p>\deg q+1$, there is no horizontal or slant asymptote; polynomial division describes the end behavior.

For example,

The canceled factor creates a hole at $x=-2$, the remaining denominator gives the vertical asymptote $x=2$, and equal degrees give the horizontal asymptote

For a slant example,

Because the remainder term approaches zero as $x$ becomes very large in magnitude, the slant asymptote is

Takeaway: Degree comparison describes end behavior, while polynomial division supplies a linear asymptote when the numerator degree is exactly one greater.

Graphing rational functions

A reliable graphing sequence is:

1. Factor the numerator and denominator.

2. State all domain restrictions.

3. Locate holes and calculate their coordinates.

4. Locate vertical asymptotes.

5. Find horizontal or slant asymptotes.

6. Find the intercepts.

7. Test points in the intervals formed by the vertical asymptotes.

8. Sketch each branch so it approaches the appropriate asymptotes.

For

the domain excludes $x=2$, the vertical asymptote is $x=2$, and the horizontal asymptote is $y=1$. The $x$-intercept is $(-1,0)$, and

so the $y$-intercept is $(0,-\frac{1}{2})$. The vertical asymptote divides the graph into the intervals $(-\\infty,2)$ and $(2,\infty)$. Testing a point from each interval helps determine the position of the branches relative to the horizontal asymptote.

A hole is represented by an open point because the function is not defined there. A vertical asymptote is not a point on the graph; it is a boundary that the branches approach.

Takeaway: A graph is most accurate when its algebraic features are found before points are plotted.

Applications of rational models

Rational models describe situations involving rates, changing denominators, or combined effects of several rates.

For distance, rate, and time,

If a car travels $120$ miles at speed $r$ and another $120$ miles at speed $r+20$, taking a total of $5$ hours, the model is

Solving gives $r=40$ or $r=-12$. Since speed must be positive, use $r=40$ miles per hour; the second speed is $60$ miles per hour. Verification gives

For combined work, rates add. If two workers complete a job in $t_1$ and $t_2$ hours, their combined time $t$ satisfies

For completion times of $6$ hours and $10$ hours,

so $t=\frac{15}{4}=3.75$ hours.

For average cost, if $F$ is fixed cost, $v$ is variable cost per unit, and $x>0$ is the number of units, then

With fixed cost $18{,}000$ and variable cost $12$ dollars per item,

At $x=600$, the average cost is

The horizontal asymptote $y=12$ represents the long-run variable cost per item.

When interpreting an application, reject values that make a denominator zero, represent a negative time or rate when positivity is required, violate a stated condition, or fail substitution into the original model. Report the final result with appropriate units.

Takeaway: Build the quotient from the quantities in the situation, solve algebraically, then filter solutions through restrictions and physical meaning.

• restriction

• least common denominator (LCD)

• extraneous solution