Whittaker Campbell
08/21/2023 · High School
Write an equation for the inverse of the given one-to-one function. \( f(x)=\frac{3 x+1}{7} \)
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Step-by-step Solution
To find the inverse of a one-to-one function, you swap the roles of x and y and then solve for y. Here's how you do it for the given function \( f(x)=\frac{3 x+1}{7} \):
1. Start with the original function:
\[ f(x) = \frac{3x + 1}{7} \]
2. Swap x and y:
\[ x = \frac{3y + 1}{7} \]
3. Solve for y:
Multiply both sides by 7 to get rid of the denominator:
\[ 7x = 3y + 1 \]
4. Subtract 1 from both sides:
\[ 7x - 1 = 3y \]
5. Divide both sides by 3 to solve for y:
\[ y = \frac{7x - 1}{3} \]
So the inverse function \( f^{-1}(x) \) is:
\[ f^{-1}(x) = \frac{7x - 1}{3} \]
Quick Answer
The inverse function is \( f^{-1}(x) = \frac{7x - 1}{3} \).
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