1. P.3 Functions & Their Graphs 1.Use function notation to represent and evaluate a function. 2.Find the domain and range of a function. 3.Sketch the graph of a function. 4.Identify different types of transformations of functions. 5.Classify functions and recognize combinations of functions.

2. Definition of a Real-Valued Function of a Real Variable Let X and Y be sets of real numbers. A real- valued function f of a real variable x from X to Y is a correspondence that assigns to each number x in X exactly one number y in Y. The domain of f is the set X. The number y is the image of x under f and is denoted by f(x), which is called the value of f at x. The range of f is a subset of Y and consists of all images of numbers in X.

3. The variable x is the independent variable and the variable y is the dependent variable. A linear equation such as Ax + By = C implicitly defines y as a function of x. Rewriting it as explicitly defines y as a function of x. In convenient function notation, replace y by f(x). A notation such as f(a) means that the function f is evaluated at x = a.

4. The Graph of a Function Memorize the graphs of these functions: 1.Identity function (linear function): y = x 2.Squaring function: y = x 2 3.Cubing function: y = x 3 4.Square-root function: 5.Absolute-value function: 6.Rational function: 7.Sine function: y = sin x 8.Cosine function: y = cos x 9.Tangent function: y = tan x

5. Transformations of y = f(x) Horizontal shift c units to the right: y = f(x – c) Horizontal shift c units to the left: y = f(x + c) Vertical shift c units downward: y = f(x) – c Vertical shift c units upward: y = f(x) + c Reflection about the x-axis: y = – f(x) Reflection about the y-axis: y = f(– x) Reflection about the origin: y = – f(– x)

6. Classification of Elementary Functions 1.Algebraic: polynomial, radical, rational 2.Trigonometric: sine, cosine, tangent, etc. 3.Exponential & Logarithmic: e x, 2 x, log a x

7. Polynomial Function n is a nonnegative integer a i are coefficients a n is the leading coefficient a 0 is the constant term If a n is not zero, then n is the degree of the polynomial function, and n may be 0, 1, 2,... Leading Coefficient Test determines the left and right behavior of the graph. f(x) = a n x n + a n-1 x n-1 + a n-2 x n-2 +... + a 1 x + a 0

8. Combination of Functions Functions such as f and g may be added, f + g, subtracted, f - g, multiplied, fg, or divided, f/g. Functions may also be combined through composition, (f o g)(x) or f(g(x)).

9. Odd & Even Functions The function y = f(x) is even if f(x) = f(– x). The function y = f(x) is odd if f(– x) = – f(x). Cosine is an even function: cos(x) = cos(– x). Sine and tangent are odd functions: sin (– x) = – sin(x) and tan (– x) = – tan(x).