Question
What is the inverse of f(x)=(12x)3?
Responses
f−1(x)=12x−−−√3
f superscript negative 1 end superscript left parenthesis x right parenthesis equals cube root of 1 half x end root
f−1(x)=2x√3
f superscript negative 1 end superscript left parenthesis x right parenthesis equals 2 cube root of x
f−1(x)=12x√3
f superscript negative 1 end superscript left parenthesis x right parenthesis equals 1 half cube root of x
f−1(x)=2x−−√3
Responses
f−1(x)=12x−−−√3
f superscript negative 1 end superscript left parenthesis x right parenthesis equals cube root of 1 half x end root
f−1(x)=2x√3
f superscript negative 1 end superscript left parenthesis x right parenthesis equals 2 cube root of x
f−1(x)=12x√3
f superscript negative 1 end superscript left parenthesis x right parenthesis equals 1 half cube root of x
f−1(x)=2x−−√3
Answers
GPT-4o mini
To find the inverse of the function \( f(x) = (12x)^3 \), we will follow the standard procedure for finding the inverse function.
1. **Start with the function definition:**
\[
y = (12x)^3
\]
2. **Swap \( x \) and \( y \):**
\[
x = (12y)^3
\]
3. **Solve for \( y \):**
- Take the cube root of both sides:
\[
\sqrt[3]{x} = 12y
\]
- Now, divide by 12:
\[
y = \frac{\sqrt[3]{x}}{12}
\]
4. **Express the inverse function:**
\[
f^{-1}(x) = \frac{\sqrt[3]{x}}{12}
\]
Now, let’s look through the options provided:
1. **Option 1:** \( f^{-1}(x) = 12x^{-\frac{1}{3}} \) (equivalent to \( \frac{12}{\sqrt[3]{x}} \))
- This does not match.
2. **Option 2:** \( f^{-1}(x) = 2\sqrt[3]{x} \)
- This does not match.
3. **Option 3:** \( f^{-1}(x) = \frac{1}{2} \sqrt[3]{x} \)
- This matches because \( \frac{\sqrt[3]{x}}{12} = \frac{1}{12}\sqrt[3]{x} \).
4. **Option 4:** \( f^{-1}(x) = 2x^{-\frac{1}{3}} \) (equivalent to \( \frac{2}{\sqrt[3]{x}} \))
- This does not match.
The only option that matches our derived inverse function is:
**Option 3:** \( f^{-1}(x) = \frac{1}{12} \sqrt[3]{x} \).
Please note that the response provided states "1/2 cube root of x" which is not correct, the correct inverse, as derived, is \( f^{-1}(x) = \frac{1}{12} \sqrt[3]{x} \).
1. **Start with the function definition:**
\[
y = (12x)^3
\]
2. **Swap \( x \) and \( y \):**
\[
x = (12y)^3
\]
3. **Solve for \( y \):**
- Take the cube root of both sides:
\[
\sqrt[3]{x} = 12y
\]
- Now, divide by 12:
\[
y = \frac{\sqrt[3]{x}}{12}
\]
4. **Express the inverse function:**
\[
f^{-1}(x) = \frac{\sqrt[3]{x}}{12}
\]
Now, let’s look through the options provided:
1. **Option 1:** \( f^{-1}(x) = 12x^{-\frac{1}{3}} \) (equivalent to \( \frac{12}{\sqrt[3]{x}} \))
- This does not match.
2. **Option 2:** \( f^{-1}(x) = 2\sqrt[3]{x} \)
- This does not match.
3. **Option 3:** \( f^{-1}(x) = \frac{1}{2} \sqrt[3]{x} \)
- This matches because \( \frac{\sqrt[3]{x}}{12} = \frac{1}{12}\sqrt[3]{x} \).
4. **Option 4:** \( f^{-1}(x) = 2x^{-\frac{1}{3}} \) (equivalent to \( \frac{2}{\sqrt[3]{x}} \))
- This does not match.
The only option that matches our derived inverse function is:
**Option 3:** \( f^{-1}(x) = \frac{1}{12} \sqrt[3]{x} \).
Please note that the response provided states "1/2 cube root of x" which is not correct, the correct inverse, as derived, is \( f^{-1}(x) = \frac{1}{12} \sqrt[3]{x} \).