For the function ​f(x)equals=4 x minus 74x−7​, determine whether​ f(x) is​ one-to-one. If​ so, write an equation for the inverse function in the form yequals=f Superscript negative 1 Baseline left parenthesis x right parenthesisf−1(x).

To determine whether the function \( f(x) = 4x - 7 \) is one-to-one, we need to check if every \( y \) value has a unique \( x \) value. This can be done by checking if the function is strictly monotonic (either strictly increasing or strictly decreasing).

The function \( f(x) = 4x - 7 \) is a linear function with a positive slope. Since the slope (4) is positive, the function is strictly increasing. A strictly increasing function is one-to-one because each \( y \) value corresponds to exactly one \( x \) value.

Since \( f(x) \) is one-to-one, we can find its inverse function. To find the inverse function, we follow these steps:

1. Write \( y = f(x) \):
\[ y = 4x - 7 \]

2. Swap \( x \) and \( y \) to get:
\[ x = 4y - 7 \]

3. Solve for \( y \) in terms of \( x \):
\[ x + 7 = 4y \]
\[ y = \frac{x + 7}{4} \]

So, the inverse function is:
\[ f^{-1}(x) = \frac{x + 7}{4} \]

Therefore, the function \( f(x) = 4x - 7 \) is one-to-one, and the inverse function is:
\[ f^{-1}(x) = \frac{x + 7}{4} \]

Would you like any more details or have any questions?

Here are 5 questions you might find interesting to ask next:
1. How can I determine if a quadratic function is one-to-one?
2. What are the steps to find the inverse of a composite function?
3. Can you explain the horizontal line test for determining if a function is one-to-one?
4. How do I verify if two functions are inverses of each other?
5. What is the difference between a one-to-one function and an onto function?

**Tip:** Always verify the domain and range of a function and its inverse to ensure they correspond correctly.

Understand how to determine if a function is one-to-one using the example of f(x) = 4x - 7. Learn to find the inverse function f^{-1}(x) = (x + 7) / 4. Suitable for high school students learning functions and inverse functions.

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