Graphing the Inverse
The horizontal line test says that if any horizontal line intersects the graph of a function at most once, then the function is
one-to-one and is invertible.
The graph of the inverse of a function is the reflection of a one-to-one f
The Law of Cosines
Like the law of sines, the law of cosines is a formula that solves for angles and sides for any given triangle, even those that
are not right triangles. It is derived from the Pythagorean theorem.
The law of cosines is used to solve
Solving a Triangle Given Two Sides and One Angle
Review: The law of sines solves for sides or angles of triangles that are not right triangles.
The law of sines:
Given two sides and one angle, use the law of sines to solve
Using Properties of Exponents to Solve Exponential Functions
Exponential equations are equations where the variable is in the exponent.
Steps for Solving an Exponential Equation By Using Properties of Exponents
1. Use properties of exponents to write ea
The Law of Sines
The law of sines allows you to solve for sides or angles of triangles that are not right triangles.
In a triangle with sides a, b, and c and opposite angles , , and , the law of sines states that
Alternatively, the law of sines is often
Finding the Inverse of a Function
To find the inverse of a function algebraically:
1. Swap the x variables with the y variables.
2. Solve the new equation for y. If this new equation cannot be solved for y as a function
of x, the original function does no
The Law of Cosines (SSS)
Review: The law of cosines has three forms. Use the form that fits the triangle:
Review: The law of cosines is used to solve a triangle when given three sides of a triangle or two sides plus the included
Graphing Exponential Functions: Useful Patterns
! Exponential functions are functions where the variable is in the exponent.
! The graphs of exponential functions like y = a x where a > 1 all have the same basic
shape. The graph grows rapidly for positive
An Introduction to Exponential Functions
Exponential functions are functions where the variable is in the exponent or power.
By now you are pretty familiar with polynomials and
polynomial functions. When dealing with polynomials,
a variable like x is rais
Understanding Inverse Functions
An inverse function undoes another function.
If g(x) is the inverse function of f(x), then g(f(x) = x and f(g(x) = x.
An inverse function reverses the original function's input and output values.
For example, if g(x) is t