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Function V Square-root and Absolute Functions By Mr Porter X-axis Y-axis y 1 = 6 -2x 6 3 √6 0 (2,-4) 4 (2, 4) X-axis Y-axis

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Square-Root Function: The important feature of this function is that f(x) must be a ZERO or POSITVE value, otherwise the function is undefined in the real number system. We will at this stage of the course, restrict f(x) to be linear functions, such as f(x) = 2x + 3 or f(x) = 4 - x. The function,, is equivalent to half of a horizontal parabola and has similar graphing features. Example Consider the function:,, squaring both sides y 2 = x – 5, re-arrange to make x the subject. x = y 2 + 5, which is the form of a horizontal parabola. The important fact in this example is that: x – 5 ≥ 0, to have any real answer. This means, the Domain is x ≥ 5 for all x in R, and the Range is y ≥ 0 for all y in R.

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Graphing : To sketch, start with y = f(x). Example 1: Sketch Hence, state clearly the functions domain and range. Now, take the square-root, top half of a parabola Remember, down here y = 2x - 5 is negative! Domain: all x ≥ 2.5 in R Range: All y ≥ 0 in R Domain is found by solving 2x - 5 ≥ 0 Solution: Find domain first, solve 6 – 2x ≥ 0 Example 2: Sketch Hence, state clearly the functions domain and range. -2x ≥ -6 x ≤ 3 This mean the curve goes to the left. Sketch y 1 = 6 - 2x on the number plane. X-axis Y-axis y 1 = 6 -2x 6 3 Domain: all x ≤ 3 in R Range: All y ≥ 0 in R X-axis Y-axis y 1 =2x-5 Sketch the graph of y = 2x - 5. √6 Now, take the square-root, top half of a parabola left.

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Harder Examples: Sketch the graph of the following function and clear state the domain and range of each. Sol: Find domain first, solve x +1 ≥ 0 x +1 ≥ 0 ==> x ≥ -1, curves to the right. Sketch y 1 = x + 1 on the same number plane and y 2 = 2. y = x+1 y = 2 x y Now, sketch the square-root curve to the right, starting at x = -1. Slide the curve up 2! Sol: Find domain first, solve 2x - 4 ≥ 0 2x – 4 ≥ 0 ==> x ≥ 2, curves to the right. Sketch y 1 = 2x – 4 on the same number plane and y 2 = 5. Domain: All x ≥ -1 in R Range: All y ≥ 2 in R Because of the (–) in-front of the square-root, the curve goes under. y 2 = 5 y 1 = 2x- 4 2 Slide the curve up 5! Domain: All x ≥ 2 in R Range: All y ≤ 5 in R

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Exercise: For each of the following: a) Sketch the function b) Define the domain and range of the function 4 2 y x Domain: all x ≤ 4 in R Range: all y ≥ 0 in R y -6x -√6 Domain: all x ≥ -6 in R Range: all y ≤ 0 in R Domain: all x ≥ -3 in R Range: all y ≥-2 in R -3x -2 5 y

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Absolute Value Functions. y = | f(x) | The absolute symbols |.. | given the value as a POSITIVE number, as in distance. Examples: Sketch and define domain and range. Step 1: Sketch f(x) = 2x – /211/2 x y Domain: all x in R Range: all y ≥ 0 Step 2: at the x-intercept, reflect graph UP. 3 Domain: all x in R Range: all y ≥ 0 (2,-4) Step 1: Sketch f(x) = x 2 - 4x 0 4 Step 2: at the x-intercepts, reflect negative part of graph UP. (2, 4)

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More Examples Step 1: Sketch f(x) =, asymptote at x = 1, y = Domain: all x in R, x ≠ 1 Range: all y in R, y > 0 Step 2: Reflect the negative part of the graph UP above the x-axis. 2 Step 1: Sketch f(x) =, asymptote at x = 1, y = Step 2: Reflect the positive part of the graph DOWN below the x-axis. -2 [An absolute graph can be NEGATIVE] Domain: all x in R, x ≠ 1 Range: all y in R, y < 0

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Exercise: For each function: i) Sketch the Functionii) Write down the domain and range. 4 2 Domain: all x in R Range: all y ≥ 0 in R Domain: all x in R, x ≠ 1.5 Range: all y ≥ 0 in R, y ≠ Domain: all x in R Range: all y ≥ 0 in R Note: f(x) = |(x - 2)(x + 2) |Note: f(x) = |(x - 6)(x + 2) | (2,-16) (2,16) Domain: all x in R Range: all y ≥ 0 in R

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