Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To find the equation of the inverse of the function [tex]\( f(x) = \left(\frac{1}{6}\right)^{x-3} \)[/tex], we need to follow a series of steps meticulously:
### Step 1: Express [tex]\( f(x) \)[/tex] as [tex]\( y \)[/tex]
Let's start by writing the function in terms of [tex]\( y \)[/tex]:
[tex]\[ y = \left(\frac{1}{6}\right)^{x-3} \][/tex]
### Step 2: Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
To find the inverse function, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \left(\frac{1}{6}\right)^{y-3} \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
Now, we need to solve this equation for [tex]\( y \)[/tex].
#### Step 3a: Take natural logarithms of both sides
Taking the natural logarithm on both sides:
[tex]\[ \ln(x) = \ln \left(\left(\frac{1}{6}\right)^{y-3}\right) \][/tex]
#### Step 3b: Use the logarithmic power rule
Using the power rule for logarithms [tex]\( \ln(a^b) = b \cdot \ln(a) \)[/tex], we get:
[tex]\[ \ln(x) = (y - 3) \cdot \ln\left(\frac{1}{6}\right) \][/tex]
#### Step 3c: Isolate [tex]\( y \)[/tex]
Next, solve for [tex]\( y \)[/tex]:
[tex]\[ y - 3 = \frac{\ln(x)}{\ln\left(\frac{1}{6}\right)} \][/tex]
Adding 3 to both sides:
[tex]\[ y = \frac{\ln(x)}{\ln\left(\frac{1}{6}\right)} + 3 \][/tex]
### Conclusion
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{\ln(x)}{\ln\left(\frac{1}{6}\right)} + 3 \][/tex]
So, the equation of the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{\ln(x)}{\ln\left(\frac{1}{6}\right)} + 3 \][/tex]
### Step 1: Express [tex]\( f(x) \)[/tex] as [tex]\( y \)[/tex]
Let's start by writing the function in terms of [tex]\( y \)[/tex]:
[tex]\[ y = \left(\frac{1}{6}\right)^{x-3} \][/tex]
### Step 2: Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
To find the inverse function, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \left(\frac{1}{6}\right)^{y-3} \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
Now, we need to solve this equation for [tex]\( y \)[/tex].
#### Step 3a: Take natural logarithms of both sides
Taking the natural logarithm on both sides:
[tex]\[ \ln(x) = \ln \left(\left(\frac{1}{6}\right)^{y-3}\right) \][/tex]
#### Step 3b: Use the logarithmic power rule
Using the power rule for logarithms [tex]\( \ln(a^b) = b \cdot \ln(a) \)[/tex], we get:
[tex]\[ \ln(x) = (y - 3) \cdot \ln\left(\frac{1}{6}\right) \][/tex]
#### Step 3c: Isolate [tex]\( y \)[/tex]
Next, solve for [tex]\( y \)[/tex]:
[tex]\[ y - 3 = \frac{\ln(x)}{\ln\left(\frac{1}{6}\right)} \][/tex]
Adding 3 to both sides:
[tex]\[ y = \frac{\ln(x)}{\ln\left(\frac{1}{6}\right)} + 3 \][/tex]
### Conclusion
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{\ln(x)}{\ln\left(\frac{1}{6}\right)} + 3 \][/tex]
So, the equation of the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{\ln(x)}{\ln\left(\frac{1}{6}\right)} + 3 \][/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.