Function - Domain & Range h k (h,k) X-axis Y-axis f(x)=(x-3)2(x+4) r – r Cubic, Circle, Hyperbola, etc By Mr Porter

Domain - independent variable Definitions Function: A function is a set of ordered pair in which no two ordered pairs have the same x-coordinate. E.g. (3,5) (2,-2) (4,5) Domain - independent variable The domain of a function is the set of all x-coordinates or first element of the ordered pairs. [the values of x for which a vertical line will cut the curve.] Range - dependent variable The range of a function is the set of all y-coordinates or the second element of the ordered pairs. [the values of y for which a horizontal line will cut the curve] Note: Students need to be able to define the domain and range from the equation of a curve or function. It is encourage that student make sketches of each function, labeling each key feature.

Cubic and Odd Power Functions. A general function can be written as f(x) = axn + bxn-1 + cxn-2 + …… + z where a ≠ 0 and n is a positive integer. If the power n is an ODD number, n = 3, 5, 7, ….. then f(x) is an odd powered function. We can generalise the domain and range as follows: Domain : All x in the real numbers, R. Range : All y in the real numbers, R. This is very true for the following functions: f(x) = 2x3 + 4 g(x) = 5 - x5 h(x) = x3 - x2 + 5x -7

Graphs of Odd Power Functions : n = 3 or 5 X-axis Y-axis f(x)=-x(x-3)(x+4) X-axis Y-axis f(x)=(x-3)2(x+4) X-axis Y-axis f(x)=(x-3)(x+4)(x+1)(x-1) Every vertical line and horizontal line will cut the curve. X-axis Y-axis f(x)=(x+3)2(x-4)3 X-axis Y-axis f(x)=2x3 Hence, Domain : all x in R Range : all y in R

Circle There are to forms of the circle: a) Standard circle centred at the origin (0,0) radius r. x2 + y2 = r2 Domain: -r ≤ x ≤ r and Range: -r ≤ y ≤ r b) General circle, centred at (h,k) with radius r. (x - h)2 + (y - k)2 = r2 Domain: -r +h ≤ x ≤ r + h and Range: -r + k ≤ y ≤ r + k All circles are RELATIONS, but by restricting the RANGE, we convert them to functions. Standard Circle 1) x2 + y2 = 9 ==> x2 + y2 = 32 2) x2 + y2 = 25 ==> x2 + y2 = 52 Circle centred (0,0), radius r = 3 units Circle centred (0,0), radius r = 5 units 3 -3 X-axis Y-axis 5 -5 X-axis Y-axis Domain: -3 ≤ x ≤ 3 Range: -3 ≤ y ≤ 3 Domain: -5 ≤ x ≤ 5 Range: -5 ≤ y ≤ 5

Hyperbola As with the circle, there are two forms of the hyperbola: a) Standard: b) General: centred (h,k) h k (h,k) Domain: All x in R, x ≠ 0 Range: All y in R, y ≠ 0 Domain: All x in R, x≠h Range: All y in R, y ≠ k The curve does not cut a VERTICAL or a HORIZONTAL line. These lines are called ASYMPTOTE lines. Example: Find the asymptotes for Hence, sketch. Y-axis X-axis The vertical asymptote is found by setting the DENOMINATOR to zero and solving for x. Vertical Asymptote. Denominator : x - 3 = 0 ==> x = 3. Horizontal Asymptote. Let x = 1 000 000, y is almost 0, i.e. y = 0 3 Asymptotes -2 3 The horizontal asymptote is a little harder to find, at this stage, use a very large value of x, say x = 1 000 000, then round off for a good common sense estimate. Domain: all x in R, x ≠ 3 Range: all y in R, y ≠ 0 The correct method is to use limits for the horizontal asymptote.

1) Find the asymptotes for Hence, sketch. 2) Find the asymptotes for Examples. 1) Find the asymptotes for Hence, sketch. 2) Find the asymptotes for Hence, sketch. Vertical Asymptote: Denominator : x - 3 = 0 ==> x = 3. Vertical Asymptote: Denominator : x - 3 = 0 ==> x = -2. Horizontal Asymptote: Let x = 1 000 000, y is almost -0, i.e. y = 0 Horizontal Asymptote: Let x = 1 000 000, y is almost -0, i.e. y = 0 Also, for , a < 0, 2nd & 4th Quadrants Also, for , a > 0, 1st & 3rd Quadrants X-axis Y-axis Y-axis X-axis 3 -2 2 3 2 Domain: all x in R, x ≠ 3 Range: all y in R, y ≠ 0 Domain: all x in R, x ≠ -2 Range: all y in R, y ≠ 0

Semi-Circles. The general form of a standard semi-circle is Represents the top half of the circle Represents the bottom half of the circle r – r – r r Domain: -r ≤ x ≤ r Range: 0 ≤ y ≤ r Domain: -r ≤ x ≤ r Range: -r ≤ y ≤ 0 To sketch s semi-circle or circle, draw the semi-circle first, then label domain and range.

Exercise. For each of the following, sketch the function (curve), then clear write down the DOMAIN and RANGE. X-axis Y-axis X-axis Y-axis -4 4 Hint: Determine if the function is a: Circle Hyperbola Semi-circle. Hint: Determine if the function is a: Circle Hyperbola Semi-circle. Domain: all x in R, x ≠ 0 Range: all y in R, y ≠ 0 Domain: -4 ≤ x ≤ 4 Range: 0 ≤ y ≤ 4 X-axis Y-axis 4 1 2 X-axis Y-axis 2√3 -2√3 Hint: Determine if the function is a: Circle Hyperbola Semi-circle. Hint: Determine if the function is a: Circle Hyperbola Semi-circle. Domain: -2√3 ≤ x ≤ -2√3 Range: -2√3 ≤ y ≤ -2√3 Domain: all x in R, x ≠ 4 Range: all y in R, y ≠ 0