Verified Answer :

  • Start by setting the given function equal to \(y\): \(y = \frac{9x - 7}{2}\).
  • Solve the equation for \(x\) by multiplying by 2, adding 7, and dividing by 9.
  • Swap the variables \(y\) and \(x\) to obtain the inverse function: \(f^{-1}(x) = \frac{2}{9}x + \frac{7}{9}\).
  • The final expression is \(\boxed{\frac{2}{9}x + \frac{7}{9}}\).

Explanation

  1. Finding the Inverse Function

To find the inverse function of \(f(x) = \frac{9x - 7}{2}\), we begin by setting the function equal to \(y\):
\[y = \frac{9x - 7}{2}\]
Our goal is to express \(x\) in terms of \(y\). First, we multiply both sides of the equation by 2 to eliminate the denominator:
\[2y = 9x - 7\]
Next, we isolate the term with \(x\) by adding 7 to both sides:
\[2y + 7 = 9x\]
Now, we divide both sides by 9 to solve for \(x\):
\[x = \frac{2y + 7}{9}\]
Distributing the division by 9 gives us:
\[x = \frac{2}{9}y + \frac{7}{9}\]
Since \(x = f^{-1}(y)\), we can write the inverse function as \(f^{-1}(y) = \frac{2}{9}y + \frac{7}{9}\). To express the inverse function in terms of \(x\), we simply swap the variable \(y\) for \(x\):
\[f^{-1}(x) = \frac{2}{9}x + \frac{7}{9}\]
Comparing this to the requested form \([?]x + \text{___}\), we identify that the missing values are \(\frac{2}{9}\) and \(\frac{7}{9}\).

Examples

The concept of inverse functions is crucial in many real-world applications, such as converting between different units. For example, if a function \(f(x)\) converts temperature from Celsius to Fahrenheit, where \(f(x) = \frac{9}{5}x + 32\), the inverse function \(f^{-1}(x)\) will convert temperature back from Fahrenheit to Celsius. Finding this inverse allows us to easily revert calculations and understand the relationship between variables in reverse.

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