Math Problem Statement

Let f ( x )

1 x + 7

f − 1 ( x )

Solution

To find the inverse function f1(x)f^{-1}(x) of the given function f(x)=1x+7f(x) = \frac{1}{x} + 7, we need to follow these steps:

Step 1: Replace f(x)f(x) with yy

Let y=f(x)=1x+7y = f(x) = \frac{1}{x} + 7.

y=1x+7y = \frac{1}{x} + 7

Step 2: Solve for xx

We want to express xx in terms of yy. First, subtract 7 from both sides:

y7=1xy - 7 = \frac{1}{x}

Next, take the reciprocal of both sides:

x=1y7x = \frac{1}{y - 7}

Step 3: Replace yy with xx

Now, replace yy with xx to get the inverse function:

f1(x)=1x7f^{-1}(x) = \frac{1}{x - 7}

Thus, the inverse function is:

f1(x)=1x7f^{-1}(x) = \frac{1}{x - 7}

Would you like more details on the process, or do you have any questions?

Here are 5 related questions to explore:

  1. What are the domain and range of the original function f(x)=1x+7f(x) = \frac{1}{x} + 7?
  2. How can we verify that f(f1(x))=xf(f^{-1}(x)) = x for this function?
  3. What are the domain and range of the inverse function f1(x)f^{-1}(x)?
  4. How does shifting a function vertically (like adding 7) affect its inverse?
  5. What happens to the graph of f(x)=1x+7f(x) = \frac{1}{x} + 7 compared to f(x)=1xf(x) = \frac{1}{x}?

Tip: When finding the inverse of a function, always check that the resulting inverse undoes the operations of the original function.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Rational Functions

Formulas

Inverse function formula f^{-1}(x)
f(x) = 1/x + 7
f^{-1}(x) = 1/(x - 7)

Theorems

The Inverse Function Theorem

Suitable Grade Level

Grades 10-12