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Essential Concepts

Rational Functions

• A rational function is a function that can be written as the quotient of two polynomial functions.
• We can use arrow notation to describe local behavior and end behavior of rational functions.
• The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero.
• A hole or removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero.
• The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.
• The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator.
  • Case 1: Degree of numerator is less than degree of denominator: horizontal asymptote at [latex]y = 0[/latex].
  • Case 2: Degree of numerator is greater than degree of denominator by one: no horizontal asymptote; slant asymptote.
    • If the degree of the numerator is greater than the degree of the denominator by more than one, the end behavior of the function’s graph will mimic that of the graph of the reduced ratio of leading terms.
  • Case 3: Degree of numerator is equal to degree of denominator: horizontal asymptote at ratio of leading coefficients.
• A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.
• Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior.
• A rational function will not have a [latex]y[/latex]-intercept if the function is not defined at zero.
• The [latex]x[/latex]-intercept(s) of a rational function can only occur when the numerator of the rational function is equal to zero.
• If a rational function has [latex]x[/latex]-intercepts at [latex]x={x}_{1},{x}_{2},\dots ,{x}_{n}[/latex], vertical asymptotes at [latex]x={v}_{1},{v}_{2},\dots ,{v}_{m}[/latex], and no [latex]{x}_{i}=\text{any }{v}_{j}[/latex], then the function can be written in the form [latex]f\left(x\right)=a\dfrac{{\left(x-{x}_{1}\right)}^{{p}_{1}}{\left(x-{x}_{2}\right)}^{{p}_{2}}\cdots {\left(x-{x}_{n}\right)}^{{p}_{n}}}{{\left(x-{v}_{1}\right)}^{{q}_{1}}{\left(x-{v}_{2}\right)}^{{q}_{2}}\cdots {\left(x-{v}_{m}\right)}^{{q}_{n}}}[/latex]

Radical Functions

• A radical function is a function that involves the use of a radical (root) symbol to indicate the root of a number or expression.
• The inverse of a quadratic function is a square root function.
• To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one.
  • For even roots (e.g., square roots), the expression inside the radical must be non-negative, as you cannot take the even root of a negative number in the real number system.
  • For odd roots (e.g., cube roots), the expression inside the radical can be any real number.
• Domain: The domain of a radical function depends on the index [latex]n[/latex]
• Range: The range of a radical function varies based on the specific function and its domain.
• To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one.
• When finding the inverse of a radical function, we need a restriction on the domain of the answer.
• Inverse and radical functions can be used to solve application problems.

Variation

• A relationship where one quantity is a constant multiplied by another quantity is called direct variation.
• Two variables that are directly proportional to one another will have a constant ratio.
• A relationship where one quantity is a constant divided by another quantity is called inverse variation.
• Two variables that are inversely proportional to one another will have a constant multiple.
• In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation.

Key Equations

Rational Function	[latex]f\left(x\right)=\dfrac{P\left(x\right)}{Q\left(x\right)}=\dfrac{{a}_{p}{x}^{p}+{a}_{p - 1}{x}^{p - 1}+...+{a}_{1}x+{a}_{0}}{{b}_{q}{x}^{q}+{b}_{q - 1}{x}^{q - 1}+...+{b}_{1}x+{b}_{0}}, Q\left(x\right)\ne 0[/latex]
Direct variation	[latex]y=k{x}^{n},k\text{ is a nonzero constant}[/latex].
Inverse variation	[latex]y=\dfrac{k}{{x}^{n}},k\text{ is a nonzero constant}[/latex].

Glossary

arrow notation

a way to symbolically represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a value

constant of variation

the non-zero value [latex]k[/latex] that helps define the relationship between variables in direct or inverse variation

direct variation

the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other

horizontal asymptote

a horizontal line [latex]y=b[/latex] where the graph approaches the line as the inputs increase or decrease without bound.

inverse variation

the relationship between two variables in which the product of the variables is a constant

inversely proportional

a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases

invertible function

any function that has an inverse function

joint variation

a relationship where a variable varies directly or inversely with multiple variables

rational function

a function that can be written as the ratio of two polynomials

removable discontinuity

a single point at which a function is undefined that, if filled in, would make the function continuous; it appears as a hole on the graph of a function

varies directly

a relationship where one quantity is a constant multiplied by the other quantity

varies inversely

a relationship where one quantity is a constant divided by the other quantity

vertical asymptote

a vertical line [latex]x=a[/latex] where the graph tends toward positive or negative infinity as the inputs approach [latex]a[/latex]