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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]
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