What is the inverse of the function f(x) = 6x + 3?
Understand the Problem
The question is asking for the inverse of the given function f(x) = 6x + 3. To solve this, we will set y = f(x), then solve for x in terms of y and express the result as the inverse function f^{-1}(x).
Answer
The inverse function is \( f^{-1}(x) = \frac{x - 3}{6} \).
Answer for screen readers
The inverse function is ( f^{-1}(x) = \frac{x - 3}{6} ).
Steps to Solve
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Set up the equation Set ( y = f(x) ). We have: $$ y = 6x + 3 $$
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Isolate x To find the inverse, we need to solve for ( x ). First, subtract 3 from both sides: $$ y - 3 = 6x $$
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Divide by the coefficient of x Next, divide both sides by 6: $$ x = \frac{y - 3}{6} $$
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Express the inverse function Now, we can express this in terms of ( x ) for the inverse function ( f^{-1}(x) ): $$ f^{-1}(x) = \frac{x - 3}{6} $$
The inverse function is ( f^{-1}(x) = \frac{x - 3}{6} ).
More Information
Finding the inverse of a function is a common task in algebra, and it allows us to reverse the relationship defined by the original function. The original function ( f(x) = 6x + 3 ) is linear, and its inverse is also linear.
Tips
- Forgetting to switch ( x ) and ( y ) when solving for the inverse.
- Incorrectly simplifying during the algebraic manipulation, such as oversight in arithmetic operations.
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