To determine whether the function $h(x) = x^2 - 7x$ is one-to-one, we need to check whether each value of $h(x)$ corresponds to exactly one value of $x$. A function is one-to-one if and only if different inputs map to different outputs, meaning that for $h(x_1) = h(x_2)$, it must follow that $x_1 = x_2$.

Step 1: Check if $h(x_1) = h(x_2)$ implies $x_1 = x_2$

Let’s set $h(x_1) = h(x_2)$:

$x_1^2 - 7x_1 = x_2^2 - 7x_2$

Rearrange the equation:

$x_1^2 - x_2^2 = 7x_1 - 7x_2$

Factor both sides:

$(x_1 - x_2)(x_1 + x_2) = 7(x_1 - x_2)$

If $x_1 \neq x_2$, we can divide both sides by $(x_1 - x_2)$ (since it is non-zero):

$x_1 + x_2 = 7$

So, if $x_1 \neq x_2$, we must have $x_1 + x_2 = 7$, which means $x_1$ and $x_2$ are not distinct values unless they satisfy this equation.

Step 2: Analyze the quadratic nature of the function

The function $h(x) = x^2 - 7x$ is a quadratic function, and quadratic functions generally fail to be one-to-one because they can have the same output for two different input values.

To further confirm this, we can find the vertex of the parabola by completing the square or using the formula for the vertex of a quadratic function $ax^2 + bx + c$:

$x_{\text{vertex}} = \frac{-b}{2a}$

For $h(x) = x^2 - 7x$, $a = 1$ and $b = -7$:

$x_{\text{vertex}} = \frac{-(-7)}{2(1)} = \frac{7}{2} = 3.5$

At $x = 3.5$, the function reaches its minimum, meaning the function decreases until $x = 3.5$ and then increases afterward. This confirms that the function is not one-to-one, as there are two values of $x$ (one on each side of the vertex) that will yield the same output.

Conclusion:

The function $h(x) = x^2 - 7x$ is not one-to-one because there are pairs of distinct values $x_1$ and $x_2$ that give the same output.

---

Would you like more details or have any other questions? Here are 5 related questions you might find interesting:

1. How can we determine the range of a quadratic function?
2. What is the significance of the vertex of a quadratic function in graphing?
3. How can we test whether a function is one-to-one using its derivative?
4. What are the conditions for a function to be one-to-one?
5. How do transformations affect whether a function is one-to-one?

Tip: For any quadratic function, if the coefficient of $x^2$ is positive, the graph opens upwards and is not one-to-one; if it is negative, the graph opens downwards and is also not one-to-one.