2Domain - independent variable DefinitionsFunction: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 variableThe domain of a function is the set of all x-coordinates or first element of theordered pairs.[the values of x for which a vertical line will cut the curve.]Range - dependent variableThe range of a function is the set of all y-coordinates or the second element of theordered 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.
3Cubic and Odd Power Functions. A general function can be written asf(x) = axn + bxn-1 + cxn-2 + …… + zwhere 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 + 4g(x) = 5 - x5h(x) = x3 - x2 + 5x -7
4Graphs of Odd Power Functions : n = 3 or 5 X-axisY-axisf(x)=-x(x-3)(x+4)X-axisY-axisf(x)=(x-3)2(x+4)X-axisY-axisf(x)=(x-3)(x+4)(x+1)(x-1)Every vertical line and horizontal line will cut the curve.X-axisY-axisf(x)=(x+3)2(x-4)3X-axisY-axisf(x)=2x3Hence,Domain : all x in RRange : all y in R
5Circle There are to forms of the circle: a) Standard circle centred at the origin (0,0) radius r.x2 + y2 = r2Domain: -r ≤ x ≤ r and Range: -r ≤ y ≤ rb) General circle, centred at (h,k) with radius r.(x - h)2 + (y - k)2 = r2Domain: -r +h ≤ x ≤ r + h and Range: -r + k ≤ y ≤ r + kAll circles are RELATIONS, but by restricting the RANGE, we convert them to functions.Standard Circle1) x2 + y2 = 9==> x2 + y2 = 322) x2 + y2 = 25==> x2 + y2 = 52Circle centred (0,0), radius r = 3 unitsCircle centred (0,0), radius r = 5 units3-3X-axisY-axis5-5X-axisY-axisDomain: -3 ≤ x ≤ 3Range: -3 ≤ y ≤ 3Domain: -5 ≤ x ≤ 5Range: -5 ≤ y ≤ 5
6Hyperbola As with the circle, there are two forms of the hyperbola: a) Standard:b) General:centred (h,k)hk(h,k)Domain:All x in R, x ≠ 0Range:All y in R, y ≠ 0Domain:All x in R, x≠hRange:All y in R, y ≠ kThe curve does not cut a VERTICAL or a HORIZONTAL line.These lines are called ASYMPTOTE lines.Example: Find the asymptotes forHence, sketch.Y-axisX-axisThe 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 = , y is almost 0, i.e. y = 03Asymptotes-23The horizontal asymptote is a little harder to find, at this stage, use a very large value of x, say x = , then round off for a good common sense estimate.Domain: all x in R, x ≠ 3Range: all y in R, y ≠ 0The correct method is to use limits for the horizontal asymptote.
71) Find the asymptotes for Hence, sketch. 2) Find the asymptotes for Examples.1) Find the asymptotes forHence, sketch.2) Find the asymptotes forHence, sketch.Vertical Asymptote: Denominator : x - 3 = 0 ==> x = 3.Vertical Asymptote: Denominator : x - 3 = 0 ==> x = -2.Horizontal Asymptote: Let x = , y is almost -0, i.e. y = 0Horizontal Asymptote: Let x = , y is almost -0, i.e. y = 0Also,for , a < 0, 2nd & 4th QuadrantsAlso,for , a > 0, 1st & 3rd QuadrantsX-axisY-axisY-axisX-axis3-2232Domain: all x in R, x ≠ 3Range: all y in R, y ≠ 0Domain: all x in R, x ≠ -2Range: all y in R, y ≠ 0
8Semi-Circles. The general form of a standard semi-circle is Represents the top halfof the circleRepresents the bottom halfof the circler– r– rrDomain: -r ≤ x ≤ rRange: 0 ≤ y ≤ rDomain: -r ≤ x ≤ rRange: -r ≤ y ≤ 0To sketch s semi-circle or circle, draw the semi-circle first, then label domain and range.
9Exercise. For each of the following, sketch the function (curve), then clear write down the DOMAIN and RANGE.X-axisY-axisX-axisY-axis-44Hint: Determine if the function is a:CircleHyperbolaSemi-circle.Hint: Determine if the function is a:CircleHyperbolaSemi-circle.Domain: all x in R, x ≠ 0Range: all y in R, y ≠ 0Domain: -4 ≤ x ≤ 4Range: 0 ≤ y ≤ 4X-axisY-axis412X-axisY-axis2√3-2√3Hint: Determine if the function is a:CircleHyperbolaSemi-circle.Hint: Determine if the function is a:CircleHyperbolaSemi-circle.Domain: -2√3 ≤ x ≤ -2√3Range: -2√3 ≤ y ≤ -2√3Domain: all x in R, x ≠ 4Range: all y in R, y ≠ 0