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Find the range of the function f(x) = 2x2 − 3x + 5 for the domain \-2, -1, 0, 1, 2\. Also, find the value of x when f(x) = -11 for the function f(x) = 3x − 2.

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Answers

2 parts
a

{4,5,7,10,19}\{4, 5, 7, 10, 19\}

b
-3

Step-by-step solution

5 steps

Here are the solutions to the problems.

Part a· 2 steps

To find the range, substitute each value from the domain into the function f(x)=2x2−3x+5f(x) = 2x^2 - 3x + 5.

  1. Calculate f(x)f(x) for each value in the domain {−2,−1,0,1,2}\{-2, -1, 0, 1, 2\}.

    For x=−2x = -2:

    f(−2)=2(−2)2−3(−2)+5=2(4)+6+5=8+6+5=19f(-2) = 2(-2)^2 - 3(-2) + 5 = 2(4) + 6 + 5 = 8 + 6 + 5 = 19

    For x=−1x = -1:

    f(−1)=2(−1)2−3(−1)+5=2(1)+3+5=2+3+5=10f(-1) = 2(-1)^2 - 3(-1) + 5 = 2(1) + 3 + 5 = 2 + 3 + 5 = 10

    For x=0x = 0:

    f(0)=2(0)2−3(0)+5=0−0+5=5f(0) = 2(0)^2 - 3(0) + 5 = 0 - 0 + 5 = 5

    For x=1x = 1:

    f(1)=2(1)2−3(1)+5=2(1)−3+5=2−3+5=4f(1) = 2(1)^2 - 3(1) + 5 = 2(1) - 3 + 5 = 2 - 3 + 5 = 4

    For x=2x = 2:

    f(2)=2(2)2−3(2)+5=2(4)−6+5=8−6+5=7f(2) = 2(2)^2 - 3(2) + 5 = 2(4) - 6 + 5 = 8 - 6 + 5 = 7

  2. Collect all the calculated f(x)f(x) values to form the range.

    The range is the set of these values, typically listed in ascending order.
    The range is {4,5,7,10,19}\boxed{\{4, 5, 7, 10, 19\}}.

Part b-3

To find the value of xx when f(x)=−11f(x) = -11 for the function f(x)=3x−2f(x) = 3x - 2.

  1. Set the function equal to −11-11.

    3x−2=−113x - 2 = -11

  2. Add 2 to both sides of the equation.

    3x=−11+23x = -11 + 2 3x=−93x = -9

  3. Divide by 3 to solve for xx.

    x=−93x = \frac{-9}{3} x=−3x = \boxed{-3}
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