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**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

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**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.

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**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

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**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

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**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

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**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 = , 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 = , 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.

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**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 = , y is almost -0, i.e. y = 0 Horizontal Asymptote: Let x = , 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

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**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.

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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

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Reciprocal Graphs Sketch and hence find the reciprocal graph y = 0 y = 1 y = 2 y = 1/2 y = 3 y = 1/3 x = 1 y = 0 Hyperbola Asymptote Domain: x R\{1}

Reciprocal Graphs Sketch and hence find the reciprocal graph y = 0 y = 1 y = 2 y = 1/2 y = 3 y = 1/3 x = 1 y = 0 Hyperbola Asymptote Domain: x R\{1}

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