At what point will the graph of f(x) = 2x - 5 and its inverse intersect?

To find the point where the graph of the function \( f(x) = 2x - 5 \) and its inverse intersect, we need to follow these steps:

### Step 1: Find the inverse of \( f(x) \)
Given \( f(x) = 2x - 5 \), we replace \( f(x) \) with \( y \):

\[
y = 2x - 5
\]

Now, solve for \( x \) in terms of \( y \):

\[
y + 5 = 2x
\]

\[
x = \frac{y + 5}{2}
\]

So, the inverse function is:

\[
f^{-1}(x) = \frac{x + 5}{2}
\]

### Step 2: Set \( f(x) \) equal to \( f^{-1}(x) \)
We set the original function equal to its inverse to find the intersection point:

\[
2x - 5 = \frac{x + 5}{2}
\]

### Step 3: Solve for \( x \)
Multiply both sides by 2 to eliminate the fraction:

\[
4x - 10 = x + 5
\]

Now, subtract \( x \) from both sides:

\[
3x - 10 = 5
\]

Add 10 to both sides:

\[
3x = 15
\]

Finally, divide by 3:

\[
x = 5
\]

### Step 4: Find the corresponding \( y \) value
Substitute \( x = 5 \) back into either \( f(x) \) or \( f^{-1}(x) \) (they should give the same result):

\[
f(5) = 2(5) - 5 = 10 - 5 = 5
\]

### Conclusion
The point where the graph of \( f(x) = 2x - 5 \) intersects with its inverse \( f^{-1}(x) = \frac{x + 5}{2} \) is \( \boxed{(5, 5)} \).

Would you like more details, or do you have any other questions?

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Here are 5 related questions you might find interesting:
1. How do you find the inverse of a quadratic function?
2. What is the significance of the intersection point of a function and its inverse?
3. Can a function and its inverse intersect more than once?
4. What happens graphically when a function is its own inverse?
5. How do you determine if a function is invertible?

**Tip:** The graphs of a function and its inverse are always symmetrical with respect to the line \( y = x \).

Discover how to find the intersection point of the function f(x) = 2x - 5 and its inverse function f^-1(x) = (x + 5) / 2. Follow a detailed step-by-step solution with equations and calculations. Suitable for high school students (Grades 10-12).

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