Notes, Lesson 1.3
New Functions from Old Functions
Definitions | Combining Functions | Check Concepts
Definitions:
 Term Definition Examples Constant Function A function for which the entire range has a constant value. , where c is a constant. Power Function A function where the independent variable is the base and constant is the exponent. , where x is the independent variable and c is the constant, and is a positive integer. Root Function An exponential function where  with n being an integer. This can also be stated as . Coefficients The numerical parts of an expression. They are usually thought of as the numbers multiplied by the variables. But constants can also be coefficients because they can be thought of as being multiplied by some variable to the zero power. 2 is the coefficient of the  term; -4 is the coefficient of the  term; and 9 is the coefficient of the  or constant term. Degree (of Polynomial) The degree of the term (monomial) which has the largest degree of each of the individual terms (monomials). This polynomial has 6 terms. The degrees of each term are as follows: term 1: degree 4 term 2: degree 3 term 3: degree 3 term 4: degree 2 term 5: degree 1 term 6: degree 0 The degree of the polynomial is 4 (the greatest of the degrees of the terms) Linear Function A linear equation is an equation whose graph is a line. A linear function is A linear equation that is also a function. Quadratic Function A quadratic equation is an equation of the form: , where a, b, and c are real numbers with a0. (The degree of the equation is 2) (The highest exponent of a variable is 2) Cubic Function A polynomial of degree 3 is of the form: Rational Function A ratio of two polynomial functions, , where . Algebraic Function A function which can be built using any algebraic operations. Rational functions all qualify, and in addition any roots can be included in the numerator and/or denominator. Trigonometric Function A function which includes algebraic operations and any of the six trigonometric definitions. (sine, cosine, tangent, cosecant, secant, cotangent) Exponential Function A function of the form , where the base a is a positive constant. Logarithmic Function A function in the form , where the base a is a positive constant. Transcendental Function A function that is not algebraic. This includes trigonometric, inverse trigonometric, exponential, and logarithmic functions. Translation A modification of a function which causes the entire function to be moved or "translated" horizonatlly from its orginal position. Depending on the value of h in the following function, f(x) will be translated left or right. If h=2, then f(x) will be translated 2 units to the right. If h=(-1), then f(x) will be translated 1 unit to the right. Stretching Transformation A modification of a function which causes the function to be "stretched out" or "shrunken" vertically or horizontally. In the function , a can "stretch" (or shrink) the sine function. If a is negative, it also causes the function to be inverted. Reflecting Transformation A modification of a function which causes the function to be reflected about the x-axis or y-axis. reflects the graph of  about the x-axis. reflects the graph of  about the y-axis.

 Using your Tools for Enriching Calculus CD (that came with your book), load and run Module 1.3. This module will let you see the effect of combining the transformations of this lesson.

Combining Functions:

 Combination Type Notation & Technique Domain of Combination Example if:  and Addition of Functions domain = Subtraction of Functions domain = Multiplication of Functions domain = Division of Functions domain = Composition of Functions domain =

 Check Concepts

 #1: What is this special function called: f(x)= -8 ? Choose One Linear Function Quadratic Function Constant Function #2: What is this special function called: f(x)= 5x ? Choose One Linear Function Quadratic Function Constant Function #3: True or False. The composition of functions is commutative. Choose One True False #4: What is this special function called:  ? Choose One Linear Function Quadratic Function Constant Function #5: True or False. The multiplication of functions is commutative. Choose One True False