Which function is the inverse of [tex]h(x)=\frac{18 x+7}{3}[/tex]?
A) [tex]k(x)=\frac{18 x-7}{3}[/tex]
B) [tex]k(x)=\frac{3 x-7}{18}[/tex]
C) [tex]k(x)=\frac{3 x+7}{18}[/tex]
D) [tex]k(x)=\frac{7 x-3}{18}[/tex]
Answer (1)
Replace h ( x ) with y : y = 3 18 x + 7 .
Swap x and y : x = 3 18 y + 7 .
Solve for y : y = 18 3 x − 7 .
Replace y with k ( x ) : k ( x ) = 18 3 x − 7 . The answer is k ( x ) = 18 3 x − 7 .
Explanation
Understanding the Problem We are given the function h ( x ) = 3 18 x + 7 and we want to find its inverse, which we'll call k ( x ) . The process for finding the inverse involves swapping x and y and then solving for y .
Replace h(x) with y First, replace h ( x ) with y : y = 3 18 x + 7
Swap x and y Next, swap x and y : x = 3 18 y + 7
Multiply by 3 Now, solve for y . Multiply both sides by 3: 3 x = 18 y + 7
Subtract 7 Subtract 7 from both sides: 3 x − 7 = 18 y
Divide by 18 Divide both sides by 18: y = 18 3 x − 7
Write the inverse function Finally, replace y with k ( x ) : k ( x ) = 18 3 x − 7 This matches option B.
Examples
In real life, inverse functions can be used to convert between different units of measurement. For example, if h ( x ) converts Celsius to Fahrenheit, then k ( x ) would convert Fahrenheit back to Celsius. Understanding inverse functions helps in reversing processes or converting back to the original state.
