Given that FX equals 2x + 8x - 10 what is the main x is less than or equal to k find the least value of k for which f is one to one

To find the least value of \( k \) for which the function \( f(x) = 2x + 8x - 10 \) is one-to-one, we need to first simplify the function and then analyze its properties.

### Step 1: Simplify the function

The given function is:

\[
f(x) = 2x + 8x - 10
\]

Combining like terms:

\[
f(x) = 10x - 10
\]

### Step 2: Analyze the function

The function \( f(x) = 10x - 10 \) is a linear function. For a linear function \( f(x) = mx + b \), where \( m \neq 0 \), the function is always one-to-one because its slope \( m \) is constant and non-zero. In our case, the slope \( m = 10 \), which is non-zero. Therefore, this function is naturally one-to-one for all values of \( x \).

### Step 3: Determine \( k \)

Since the function is already one-to-one for all \( x \), the least value of \( k \) for which \( f(x) \) is one-to-one is simply \( k = \infty \). This means the function is one-to-one on the entire real line, and there is no need to restrict \( x \) to be less than or equal to any specific value \( k \).

Thus, the least value of \( k \) is:

\[
\boxed{k = \infty}
\]

### Questions for further exploration:

1. What is the general form of a one-to-one function?
2. How do you find the inverse of a linear function like \( f(x) = 10x - 10 \)?
3. What is the significance of the slope in determining if a function is one-to-one?
4. What happens if the slope \( m \) is zero? Can the function still be one-to-one?
5. How does the concept of one-to-one relate to the horizontal line test?

### Tip:
To quickly check if a function is one-to-one, you can use the horizontal line test. If any horizontal line intersects the graph of the function at most once, the function is one-to-one.

Given that f(x) = 2x + 8x - 10, what is the least value of k for which f is one-to-one, if x is less than or equal to k?

This detailed solution explains how to find the least value of k for which the linear function f(x) = 10x - 10 is one-to-one. Learn about the one-to-one property of linear functions and the significance of the slope in determining one-to-one behavior. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation