A3. Functions 1. Intervals 2. What is a function? 3. Graphs of functions 4. Inverse functions 5. Trigonometric and inverse trigonometric functions 6. Exponential and logarithmic functions

1. Intervals Type of intervalsInequality formInterval form Closed intervalsa ≤ x ≤ b[a, b] Open intervalsa < x < b(a, b) Half open intervalsa ≤ x < b[a, b) a < x ≤ b(a, b] Infinite intervalsx ≤ b(-∞, b] x > a(a, ∞)

2. Functions Definition: A function f is a rule that assigns to each value of a real variable x exactly one value of another real variable y. The variable x is called the independent variable and the variable y is called the dependent variable. The set of values for which f is defined is called the domain of the function. The set of values that y are assigned is called the range of f. We usually write y = f(x) to express the fact that y is a function of x. Here f(x) is the name of the function.

DomainRangey = f(x) xy e.g. Let f(x) = x 2 + 2x + 4. So f(0) = 4, f(1) = 7, f(2) = 12.

Example: Specify the domains the following functions:

Example:

3. Graphs of functions  Linear functions: f(x) = mx + c  Quadratic functions: f(x) = ax 2 + bx + c  Reciprocal function: f(x) = 1/x x y f(x) = x + 3 x y f(x) = x 2 x y f(x) = 1/x

 Modulus (or absolute value) function: x y = -x y = x y

Example: Solve

The graph of y = f(x) + k is the graph of y = f(x) moved up k unites (down if k is negative). The graph of y = f(x - k) is the graph of y = f(x) moved to the right k unites (to the left if k is negative). Multiplying a function by a constant c stretches the graph vertically (if c > 1) or shrinks the graph vertically (if 0 < c < 1). A negative sign reflects the graph about the x-axis.

Example: Sketch the graphs of (i)(i) (0,-4) (1,0) (ii) (-1,4) (iii) (1,2) (iv) (v)(v) (-2,5)

4. Inverse functions Definitions: A function f is said to be increasing on its domain if f(x 1 ) < f(x 2 ) whenever x 1 < x 2. A function f is said to be decreasing on its domain if f(x 1 ) > f(x 2 ) whenever x 1 < x 2. xx y y x1x1 x1x1 x2x2 x2x2 f(x 1 ) f(x 2 )

A function f with domain D f and range R f is called one-to-one if whenever x 1 ≠ x 2 in D f, then f (x 1 ) ≠ f(x 2 ) in R f. A function f is one-to-one if and only if no horizontal line intersects its graph more than once. A function f is one-to-one on D f if it is always increasing (or decreasing) on D f. Example: Show that f(x) = 3x + 4 is one-to-one.

To find the inverse function f -1 of a function f: 1. Write the equation form y = f(x). 2. Solve the equation for x in terms of y. 3. interchange x and y. Example: Find the inverse function of f(x) = 2x – 3.

The inverse of The graph of f and f -1 are symmetrical with respect to the line y = x. x y

Example: Let f(x) = x 2 + 2x – 3. (i)Find the domain and the range of f. (ii)Show that f is not one-to-one and determine a sub- domain in which f is one-to-one. (iii) Find f -1 and state its domain and range. (-1,-4)

5. Trigonometric and inverse trigonometric functions A function f is said to be periodic, with period p, if f(x + p) = f(x) for all x in the domain of f. The sine function: f(x) = sin x Period = 2π. Domain = R, Range = [-1,1]. x y 1

The cosine function: f(x) = cos x Period = 2π. Domain = R, Range = [-1,1]. The tangent function: f(x) = tan x Period = π Domain = R\{(2n+1)π/2}, Range R x y 1 x y

Let f(x) = sin x, –π/2 ≤ x ≤ π/2. It is clear that f is one-to-one and the range of f is [-1,1]. The function arcsin(x) with domain [-1,1] and range [–π/2,π/2] is defined to be the inverse sine function. e.g. arcsin(1/2) = π/6, arcsin(1) = π/2, arcsin(sin x) = x, sin(arcsin x) = x. If y = sin x, then x = arcsin(y) and If y = arcsin x, then x = sin(y). The function arccos(x), with domain [-1,1] and range [0,π] is defined to be the inverse cosine function. The function arctan(x), with domain (-∞,∞) and range [–π/2,π/2] is defined to be the inverse tangent function.

6. Exponential and logarithmic functions Definition: The function f(x) = a x, where a is a positive real number, is called an exponential function with base a. e.g. f(x) = 2 x, g(x) = (1/2) x = 2 -x x 2x2x 1 y 2 -x

Definition: log a x is defined to be the power of a needed to get x. e.g. log = 3 because 10 3 = 1000, log 2 16 = 4 because 2 4 = 16. y = log a x is the logarithmic function with base a. If y = a x, then x = log a y; i.e. log a a x = x. If y = log a x, then x = a y. log a x is defined only if x and b ≠ 1 are both positives. log e x = ln x is called the natural logarithmic function. log 10 x is called the common logarithmic function.

Example: Solve for x the following equations:

Example: Solve the equation

Example: Let f(x) = ln(2x+3) +2. a)Find the domain of f. b)Find the inverse function of f. c)State the domain and the range of f -1. B1