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The input and output values for two functions, f of x equals 8 x Superscript 3 Baseline and g of x equals 2 Superscript x Baseline, are shown in the table.



x 0 0.25 0.5 1 2 3 4 5 10 15 20
f of x 0 0.125 1 8 64 216 512 1,000 8,000 27,000 64,000
g of x 1 1.19 1.41 2 4 8 16 32 1,024 32,768 1,048,576

Question
Over which intervals shown in the table is the function g of x growing at a faster rate than the function f of x? Select ,begin emphasis,all,end emphasis, the correct intervals.

Answer options with 5 options
A.
0 to 0.25

B.
0 to 0.5

C.
0 to 20

D.
1 to 10

E.
15 to 20

1 answer

To determine the intervals where the function \( g(x) \) is growing at a faster rate than \( f(x) \), we can compare the outputs of both functions at the given \( x \) values.

Values from the Functions:

  • For \( f(x) = 8x^3 \):

    • \( f(0) = 0 \)
    • \( f(0.25) = 0.125 \)
    • \( f(0.5) = 1 \)
    • \( f(1) = 8 \)
    • \( f(2) = 64 \)
    • \( f(3) = 216 \)
    • \( f(4) = 512 \)
    • \( f(5) = 1000 \)
    • \( f(10) = 8000 \)
    • \( f(15) = 27000 \)
    • \( f(20) = 64000 \)

  • For \( g(x) = 2^x \):

    • \( g(0) = 1 \)
    • \( g(0.25) = 1.19 \)
    • \( g(0.5) = 1.41 \)
    • \( g(1) = 2 \)
    • \( g(2) = 4 \)
    • \( g(3) = 8 \)
    • \( g(4) = 16 \)
    • \( g(5) = 32 \)
    • \( g(10) = 1024 \)
    • \( g(15) = 32768 \)
    • \( g(20) = 1048576 \)

Comparing Growth Rates in Each Interval:

  • Interval 0 to 0.25:

    • \( f(0) = 0 \), \( f(0.25) = 0.125 \)
    • \( g(0) = 1 \), \( g(0.25) = 1.19 \)
    • \( g(x) > f(x) \): True

  • Interval 0 to 0.5:

    • \( f(0.5) = 1 \)
    • \( g(0.5) = 1.41 \)
    • \( g(x) > f(x) \): True

  • Interval 0 to 20:

    • As shown, \( g(x) > f(x) \) from \( 0 \) to \( 20 \): True

  • Interval 1 to 10:

    • \( f(1) = 8 \), \( g(1) = 2 \)
    • \( g(2) = 4 \), \( f(2) = 64 \)
    • \( f(3) = 216 \), \( g(3) = 8 \)
    • \( f(4) = 512 \), \( g(4) = 16 \)
    • \( f(5) = 1000 \), \( g(5) = 32 \)
    • At \( x = 1, 2, 3, 4, 5 \): \( f(x) > g(x) \): False

  • Interval 15 to 20:

    • \( f(15) = 27000 \), \( g(15) = 32768 \)
    • \( f(20) = 64000 \), \( g(20) = 1048576 \)
    • \( g(x) > f(x) \) at first but then grows significantly faster at \( g(20) >> f(20) \): True

Conclusion:

The intervals where \( g(x) \) grows at a faster rate than \( f(x) \) are:

Correct options:

  • A. 0 to 0.25
  • B. 0 to 0.5
  • C. 0 to 20
  • E. 15 to 20

Thus, the answer options are A, B, C, and E.