# 7.7 – Solving Rational Functions

## Key Terms

• Asymptote – A line on a graph that a curve approaches more and more closely without intersecting.
• A line that a function gets closer and closer to but does not reach (or cross).
• Domain – The set of all allowable input values.
• If the function is graphed, the domain is the set of all x-values on the graph.
• Horizontal Asymptote – A horizontal line that the graph of a function approaches but never intersects.
• A function has a horizontal asymptote at every y-value where it is undefined and near which the function’s values become very large positive or negative numbers.
• Range – The set of a function’s output values (or y-values).
• Rational Function – A function that is written in terms of a rational expression.
• Vertical Asymptote – A vertical line that the graph of a function approaches but never intersects.
• A function has a vertical asymptote at every x-value where it is undefined and near which the function’s values become very large positive or negative numbers.

## Notes

Rational Functions
• A rational function is a function whose equation contains a rational expression.
• If the equation of a function is a rational expression, the function is rational.
• Example 1: $f(x)=\frac{6x-3}{4}$
• Example 2:

• The width can never be zero because you cannot divide by zero (undefined); so, domain: w > 0.
• The height cannot be zero because the rectangle would not exist; so, range: H(w) > 0
• Height and width must never be negative, because they are both real-world lengths.

Direct Variation Inverse Variation
• If you double the input of a function and it results in four times the output, and if you triple the input and it results in six times the output, then the function is most likely directly proportional because more input results in more output.
• Ex. In the direct variation function, when the function’s input value is multiplied by 4, the output is multiplied by 4.
• Ex. In the direct variation function, when the function’s input value is divided by 6, the output is divided by 6.
• If you double the input of a function and it results in half the output, and if you triple the input and it results in a third of the output, then the function is most likely inversely proportional because more input results in less output.
• Ex. In the inverse variation function, when the function’s input value is multiplied by 4, the output is divided by 4.
• Ex. In the inverse variation function, when the function’s input value is divided by 6, the output is multiplied by 6.
Asymptotes
• Horizontal Asymptote
• On the graph of H(w) below, the w-axis is also the horizontal asymptote for H(w).

• Vertical Asympotote
• On the graph of H(w) below, the line w = 0 is also the vertical asymptote for H(w).

## Examples

 Ex 1. What restrictions are there on the range of the function H(w) below? Answer: H(w) > 0 Ex 2. According to the graph of H(w) below, what happens when w gets very large? Answer: H(w) gets very small. Ex 3. According to the graph of H(w) below, what happens when w gets close to zero? Answer: H(w) gets very large.