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    Mathematics
    Grade10
    Easy

    Question

    Write a piecewise-defined function for the graph below

    1. f(x) : {(3.5x – 8 , x ≥ 2) , (2x – 3 , x < 2)}
    2. f(x) : {(6x + 7, x ≥ 2) , (-0.6x + 0.8 , x < 2)}
    3. f(x) : {(3.5x – 9, x ≥ 2) , (0.5x – 0.8 , x < 2)}
    4. f(x) : {(2.5x – 7, x ≥ 2) , (-0.6x – 0.8 , x < 2)}

    hintHint:

    In this question, we have to find the piecewise defined function for the given graph i.e., a function in which more than one formula is used to define the output over different pieces of the domain. Firstly, we will consider x ≥ 2 and using the points (4, 3) and (2, -2) to the find the equation of the line. Later, we will consider x< 2 and (-3, 1) and (2, -2) to find the other equation. The function two-equation obtained is the required answer.

    The correct answer is: f(x) : {(2.5x – 7, x ≥ 2) , (-0.6x – 0.8 , x < 2)}

      For x ≥ 2, let’s find the equation of the graph:
      General equation of a line: y = mx + c
      Consider the points (4, 3) and (2, -2).
      m = fraction numerator left parenthesis negative 2 comma space minus 3 right parenthesis over denominator left parenthesis 2 comma space minus 4 right parenthesis end fraction = fraction numerator negative 5 over denominator negative 2 end fraction = 5 over 2 = 2.5
      3 = 2.5 × 4 + c (Putting (4, 3) values in a general equation)
      c = 3 – 10 = -7
      Equation: y = 2.5x – 7 for x ≥ 2
      For x < 2, let’s find the equation of the graph:
      General equation of a line: y = mx + c
      Consider the points (-3, 1) and (2, -2).
      m = (-2 – 1)/ (2 – (-3)) = -3 over 5 = -0.6
      -2 = -0.6 × 2 + c (Putting (2, -2) values in general equation)
      c = -2 + 1.2 = -0.8
      Equation: y = -0.6x – 0.8 for x < 2
      f(x): {(2.5x – 7, x ≥ 2), (-0.6x – 0.8, x < 2)}

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