Presentation on theme: "Function - Domain & Range"— Presentation transcript:
1 Function - Domain & Range hk(h,k)X-axisY-axisf(x)=(x-3)2(x+4)r– rCubic, Circle, Hyperbola, etcBy Mr Porter
2 Domain - 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.
3 Cubic 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
4 Graphs 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
5 Circle 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
6 Hyperbola 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.
7 1) 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
8 Semi-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.
9 Exercise. 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