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.