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Given the functions F(x) and f(x), evaluate parts of them. F(x) is a piecewise function and f(x) is a quadratic.

Mathematics

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Step-by-step solution

3 steps

Step 1: Compute F(−3)F(-3).
The function is defined as:

F(x)={x2if x>1x−4if x≤1F(x) = \begin{cases} x^2 & \text{if } x > 1 \\ x - 4 & \text{if } x \le 1 \end{cases}

For F(−3)F(-3), since −3≤1-3 \le 1, we use the second rule F(x)=x−4F(x) = x - 4.

F(−3)=−3−4=−7F(-3) = -3 - 4 = -7

Step 2: Compute F(5)F(\sqrt{5}).
For F(5)F(\sqrt{5}), we need to compare 5\sqrt{5} with 1.
Since 5≈2.236\sqrt{5} \approx 2.236, which is greater than 1, we use the first rule F(x)=x2F(x) = x^2.

F(5)=(5)2=5F(\sqrt{5}) = (\sqrt{5})^2 = 5

The computed values are:
F(−3)=−7F(-3) = \boxed{-7}
F(5)=5F(\sqrt{5}) = \boxed{5}

Part b· 1 step

Given the function f(x)=−x2+4x−5f(x) = -x^2 + 4x - 5. This is a quadratic function in the form ax2+bx+cax^2 + bx + c, where a=−1a = -1, b=4b = 4, and c=−5c = -5.

  1. Find the line of symmetry.

    The formula for the line of symmetry of a quadratic function is x=−b2ax = -\frac{b}{2a}.

    x=−42(−1)=−4−2=2x = -\frac{4}{2(-1)} = -\frac{4}{-2} = 2

Part b(i)· 1 step

The line of symmetry is* x=2\boxed{x = 2}.

  1. Find the turning point.

    The xx-coordinate of the turning point is the line of symmetry, x=2x = 2.
    To find the yy-coordinate, substitute x=2x = 2 into the function f(x)f(x).

    f(2)=−(2)2+4(2)−5f(2) = -(2)^2 + 4(2) - 5 f(2)=−4+8−5f(2) = -4 + 8 - 5 f(2)=4−5=−1f(2) = 4 - 5 = -1

Part b(ii)· 1 step

The turning point is* (2,−1)\boxed{(2, -1)}.

  1. Find the maximum or minimum value of f(x)f(x).

    Since the coefficient a=−1a = -1 is negative, the parabola opens downwards, which means the turning point is a maximum point. The maximum value of the function is the yy-coordinate of the turning point.

Part b(iii)

The maximum value of f(x)f(x) is* −1\boxed{-1}.

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