Question
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
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
Answers
GPT-4o mini
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**.
\[
f(x) = A \cdot b^x
\]
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**.