Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
Let's analyze the given functions and their ranges step-by-step to determine which one has the same range as [tex]\( f(x) = -\frac{5}{7}\left(\frac{3}{5}\right)^x \)[/tex].
### Step 1: Analyzing the Range of [tex]\( f(x) \)[/tex]
The function [tex]\( f(x) = -\frac{5}{7}\left(\frac{3}{5}\right)^x \)[/tex] has a leading negative sign, which means its range will always be negative because:
- The base [tex]\(\left(\frac{3}{5}\right)^x\)[/tex] is a positive number for any real [tex]\( x \)[/tex].
- Multiplying this positive value by the negative [tex]\(-\frac{5}{7}\)[/tex] results in a negative value.
Hence, the range of [tex]\( f(x) \)[/tex] is negative.
### Step 2: Analyzing Each Candidate Function [tex]\( g(x) \)[/tex]
#### Function [tex]\( g_1(x) = \frac{5}{7}\left(\frac{3}{5}\right)^{-x} \)[/tex]
- The base [tex]\(\left(\frac{3}{5}\right)^{-x}\)[/tex] is positive for any real [tex]\( x \)[/tex] since it represents the reciprocal of [tex]\(\left(\frac{3}{5}\right)\)[/tex].
- Multiplying this positive value by the positive [tex]\(\frac{5}{7}\)[/tex] results in a positive value.
Hence, the range of [tex]\( g_1(x) \)[/tex] is positive.
#### Function [tex]\( g_2(x) = -\frac{5}{7}\left(\frac{3}{5}\right)^{-x} \)[/tex]
- The base [tex]\(\left(\frac{3}{5}\right)^{-x}\)[/tex] is positive for any real [tex]\( x \)[/tex].
- Multiplying this positive value by the negative [tex]\(-\frac{5}{7}\)[/tex] results in a negative value.
Hence, the range of [tex]\( g_2(x) \)[/tex] is negative.
#### Function [tex]\( g_3(x) = \frac{5}{7}\left(\frac{3}{5}\right)^x \)[/tex]
- The base [tex]\(\left(\frac{3}{5}\right)^x\)[/tex] is positive for any real [tex]\( x \)[/tex].
- Multiplying this positive value by the positive [tex]\(\frac{5}{7}\)[/tex] results in a positive value.
Hence, the range of [tex]\( g_3(x) \)[/tex] is positive.
#### Function [tex]\( g_4(x) = -\left(-\frac{5}{7}\right)\left(\frac{5}{3}\right)^x \)[/tex]
- Simplifying [tex]\(-\left(-\frac{5}{7}\right)\)[/tex], we get [tex]\(\frac{5}{7}\left(\frac{5}{3}\right)^x\)[/tex].
- The base [tex]\(\left(\frac{5}{3}\right)^x\)[/tex] is positive for any real [tex]\( x \)[/tex].
- Multiplying this positive value by the positive [tex]\(\frac{5}{7}\)[/tex] results in a positive value.
Hence, the range of [tex]\( g_4(x) \)[/tex] is positive.
### Conclusion
After analyzing the ranges of all functions, we find that only [tex]\( g_2(x) = -\frac{5}{7}\left(\frac{3}{5}\right)^{-x} \)[/tex] has a negative range, matching the range of [tex]\( f(x) \)[/tex]. Therefore, the function that has the same range as [tex]\( f(x) = -\frac{5}{7}\left(\frac{3}{5}\right)^x \)[/tex] is:
[tex]\[ g(x) = -\frac{5}{7}\left(\frac{3}{5}\right)^{-x} \][/tex]
### Step 1: Analyzing the Range of [tex]\( f(x) \)[/tex]
The function [tex]\( f(x) = -\frac{5}{7}\left(\frac{3}{5}\right)^x \)[/tex] has a leading negative sign, which means its range will always be negative because:
- The base [tex]\(\left(\frac{3}{5}\right)^x\)[/tex] is a positive number for any real [tex]\( x \)[/tex].
- Multiplying this positive value by the negative [tex]\(-\frac{5}{7}\)[/tex] results in a negative value.
Hence, the range of [tex]\( f(x) \)[/tex] is negative.
### Step 2: Analyzing Each Candidate Function [tex]\( g(x) \)[/tex]
#### Function [tex]\( g_1(x) = \frac{5}{7}\left(\frac{3}{5}\right)^{-x} \)[/tex]
- The base [tex]\(\left(\frac{3}{5}\right)^{-x}\)[/tex] is positive for any real [tex]\( x \)[/tex] since it represents the reciprocal of [tex]\(\left(\frac{3}{5}\right)\)[/tex].
- Multiplying this positive value by the positive [tex]\(\frac{5}{7}\)[/tex] results in a positive value.
Hence, the range of [tex]\( g_1(x) \)[/tex] is positive.
#### Function [tex]\( g_2(x) = -\frac{5}{7}\left(\frac{3}{5}\right)^{-x} \)[/tex]
- The base [tex]\(\left(\frac{3}{5}\right)^{-x}\)[/tex] is positive for any real [tex]\( x \)[/tex].
- Multiplying this positive value by the negative [tex]\(-\frac{5}{7}\)[/tex] results in a negative value.
Hence, the range of [tex]\( g_2(x) \)[/tex] is negative.
#### Function [tex]\( g_3(x) = \frac{5}{7}\left(\frac{3}{5}\right)^x \)[/tex]
- The base [tex]\(\left(\frac{3}{5}\right)^x\)[/tex] is positive for any real [tex]\( x \)[/tex].
- Multiplying this positive value by the positive [tex]\(\frac{5}{7}\)[/tex] results in a positive value.
Hence, the range of [tex]\( g_3(x) \)[/tex] is positive.
#### Function [tex]\( g_4(x) = -\left(-\frac{5}{7}\right)\left(\frac{5}{3}\right)^x \)[/tex]
- Simplifying [tex]\(-\left(-\frac{5}{7}\right)\)[/tex], we get [tex]\(\frac{5}{7}\left(\frac{5}{3}\right)^x\)[/tex].
- The base [tex]\(\left(\frac{5}{3}\right)^x\)[/tex] is positive for any real [tex]\( x \)[/tex].
- Multiplying this positive value by the positive [tex]\(\frac{5}{7}\)[/tex] results in a positive value.
Hence, the range of [tex]\( g_4(x) \)[/tex] is positive.
### Conclusion
After analyzing the ranges of all functions, we find that only [tex]\( g_2(x) = -\frac{5}{7}\left(\frac{3}{5}\right)^{-x} \)[/tex] has a negative range, matching the range of [tex]\( f(x) \)[/tex]. Therefore, the function that has the same range as [tex]\( f(x) = -\frac{5}{7}\left(\frac{3}{5}\right)^x \)[/tex] is:
[tex]\[ g(x) = -\frac{5}{7}\left(\frac{3}{5}\right)^{-x} \][/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
