Practice Problems

Answers

  1. Function g is a reflection of function f. There are different kinds of function reflections:

    Expression Type Meaning
    -f(x)

    Vertical (reflection across the x-axis)

    Every point on the graph of y=f(x) corresponds to a point on the graph of ‍y=-f(x) with the same ‍x-value but the opposite‍ y-value.
    f(-x)

    Horizontal (reflection across the y-axis)

    Every point on the graph of y=f(x) corresponds to a point on the graph of ‍y=f(-x) with the same ‍y-value but the opposite x-value.
    -f(-x)

    Vertical and horizontal

    Every point on the graph of y=f(x) corresponds to a point on the graph of ‍y=-f(-x) with the opposite ‍x- and y-values.

    We need to determine which reflection to apply on f to get ‍g, and then we need to find the expression for ‍g.

    g is a horizontal reflection of f.

    So g(x)=f(-x).

    Now recall that f(x)=-2\log_2(x+1)+3. Let's see what expression we get for f(-x):

    \begin{aligned}f(-x)&=-2\log_2\big((-x)+1\big)+3\\\\&=-2\log_2(1-x)+3\end{aligned}

    In conclusion, g(x)=-2\log_2(1-x)+3.

  2. Function g is a reflection of function f. There are different kinds of function reflections:

    Expression Type Meaning
    -f(x)

    Vertical (reflection across the x-axis)

    Every point on the graph of y=f(x) corresponds to a point on the graph of ‍y=-f(x) with the same ‍x-value but the opposite‍ y-value.
    f(-x)

    Horizontal (reflection across the y-axis)

    Every point on the graph of y=f(x) corresponds to a point on the graph of ‍y=f(-x) with the same ‍y-value but the opposite x-value.
    -f(-x)

    Vertical and horizontal

    Every point on the graph of y=f(x) corresponds to a point on the graph of ‍y=-f(-x) with the opposite ‍x- and y-values.

    First, we need to determine which kind of reflection is ‍g(x)=-\sqrt[3]{x-2}.

    Recall that f(x)=\sqrt[3]{x-2} Let's see what expression we get for  -f(x):

    -f(x)=-\sqrt[3]{x-2}

    So g(x)=-f(x).

    Since g(x)=-f(x), it is a vertical reflection of ‍f.

    So the correct answer is B.

    q2-answer-b

  3. Function g is a reflection of function f. There are different kinds of function reflections:

    Expression Type Meaning
    -f(x)

    Vertical (reflection across the x-axis)

    Every point on the graph of y=f(x) corresponds to a point on the graph of ‍y=-f(x) with the same ‍x-value but the opposite‍ y-value.
    f(-x)

    Horizontal (reflection across the y-axis)

    Every point on the graph of y=f(x) corresponds to a point on the graph of ‍y=f(-x) with the same ‍y-value but the opposite x-value.
    -f(-x)

    Vertical and horizontal

    Every point on the graph of y=f(x) corresponds to a point on the graph of ‍y=-f(-x) with the opposite ‍x- and y-values.

    g is a horizontal reflection of f.

    So g(x)=f(-x).

  4. Function g is a reflection of function f. There are different kinds of function reflections:

    Expression Type Meaning
    -f(x)

    Vertical (reflection across the x-axis)

    Every point on the graph of y=f(x) corresponds to a point on the graph of ‍y=-f(x) with the same ‍x-value but the opposite‍ y-value.
    f(-x)

    Horizontal (reflection across the y-axis)

    Every point on the graph of y=f(x) corresponds to a point on the graph of ‍y=f(-x) with the same ‍y-value but the opposite x-value.
    -f(-x)

    Vertical and horizontal

    Every point on the graph of y=f(x) corresponds to a point on the graph of ‍y=-f(-x) with the opposite ‍x- and y-values.

    Since g(x)=f(-x), it is a horizontal reflection of ‍f.

    So the correct answer is C.

    q4-answer-c
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