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Choose ,begin emphasis,all,end emphasis, the functions which decrease at a rate of 40% per unit increase in x.

Answer options with 6 options
A.
f of x equals three-halves left-parenthesis two-fifths right-parenthesis Superscript x Baseline

B.
f of x equals two-thirds left-parenthesis three-fifths right-parenthesis Superscript x Baseline

C.
f of x equals two-fifths left-parenthesis five-eighths right-parenthesis Superscript x Baseline

D.
f of x equals 1.75 left-parenthesis 0.6 right-parenthesis Superscript x Baseline

E.
f of x equals 2.5 left-parenthesis 0.4 right-parenthesis Superscript x Baseline

F.
f of x equals 0.4 left-parenthesis 0.8 right-parenthesis Superscript x Baseline

1 answer

To determine which functions decrease at a rate of 40% per unit increase in \( x \), we need to look for exponential functions of the form:

\[ f(x) = A \cdot b^x \]

where the base \( b \) must be less than 1 because we want a decreasing function. A rate of decrease of 40% means that for every unit increase in \( x \), the function value should be multiplied by \( 1 - 0.4 = 0.6 \). Thus, we want:

\[ b = 0.6 \]

Now, let's evaluate each option to see if the base matches 0.6:

A. \( f(x) = \frac{3}{2} \left(\frac{2}{5}\right)^x \)

  • Base: \( \frac{2}{5} = 0.4 \) (decreasing but not at 40%)

B. \( f(x) = \frac{2}{3} \left(\frac{3}{5}\right)^x \)

  • Base: \( \frac{3}{5} = 0.6 \) (decreases at 40%)

C. \( f(x) = \frac{2}{5} \left(\frac{5}{8}\right)^x \)

  • Base: \( \frac{5}{8} = 0.625 \) (decreasing but not at 40%)

D. \( f(x) = 1.75 (0.6)^x \)

  • Base: \( 0.6 \) (decreases at 40%)

E. \( f(x) = 2.5 (0.4)^x \)

  • Base: \( 0.4 \) (decreasing but not at 40%)

F. \( f(x) = 0.4 (0.8)^x \)

  • Base: \( 0.8 \) (decreasing but not at 40%)

The functions that decrease at a rate of 40% per unit increase in \( x \) are:

B. \( f(x) = \frac{2}{3} \left(\frac{3}{5}\right)^x \)
D. \( f(x) = 1.75 (0.6)^x \)

So the final answer is B and D.