Problem 11 - Algebra and Trigonometry

Question

Determine whether the function is one-to-one. \(h(x)=x^{2}-2 x\)

Step-by-Step Solution

Answer

The function \( h(x) = x^2 - 2x \) is not one-to-one.

Step 1: Understand the Definition of a One-to-One Function

A function is considered one-to-one if every output value is associated with exactly one input value. In mathematical terms, a function \( f(x) \) is one-to-one if \( f(a) = f(b) \) implies that \( a = b \).

Step 2: Analyze the Derivative

One way to determine if a function is one-to-one is to check the sign of its derivative. If the derivative does not change sign (i.e., is either always positive or always negative), the function is one-to-one. Calculate the derivative of \( h(x) = x^2 - 2x \):\[ h'(x) = \frac{d}{dx}(x^2 - 2x) = 2x - 2 \]

Step 3: Analyze the Sign of the Derivative

Examine the sign of \( h'(x) = 2x - 2 \). Set the derivative equal to zero to find critical points:\[ 2x - 2 = 0 \]\[ 2x = 2 \]\[ x = 1 \]The derivative changes sign at \( x = 1 \). This indicates that the function \( h(x) \) is not consistently increasing or decreasing, meaning it's not one-to-one.

Step 4: Conclude Using Horizontal Line Test

The Horizontal Line Test states that if any horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one. For \( h(x) = x^2 - 2x \), the graph is a parabola, opening upwards, which implies horizontal lines will intersect it at two points. This also confirms the function is not one-to-one.

Key Concepts

Derivative

When discussing one-to-one functions, the derivative offers a vital piece of information to understand if a function is one-to-one. The derivative of a function is essentially the slope of the function at any given point. In mathematical terms, if we have a function \( h(x) = x^2 - 2x \), its derivative, calculated as \( h'(x) = 2x - 2 \), tells us how the function values change as \( x \) changes.

• If a function's derivative is always positive, the function is continuously increasing.
• If the derivative is always negative, the function is continuously decreasing.

When a function consistently increases or decreases, it is more likely to be one-to-one, because it does not repeat output values for different input values. But, if the derivative changes sign, as it does here (from positive to negative or vice versa), the function may not be one-to-one. In the case of \( h(x) \), the derivative \( 2x - 2 \) changes sign at \( x = 1 \), suggesting that \( h(x) \) is neither always increasing nor always decreasing, indicating that it is not one-to-one.

Horizontal Line Test

The Horizontal Line Test is a visual method to determine if a function is one-to-one. The principle is simple:

• Draw or imagine a horizontal line across the graph of the function.
• If the horizontal line crosses the graph at more than one point, the function is not one-to-one.

For the function \( h(x) = x^2 - 2x \), if we plot its graph, we can see a parabola opening upwards. A horizontal line passing through this parabola can intersect it at two distinct points. This visual cue confirms that there are multiple inputs for the same output, reinforcing the notion that \( h(x) \) is not one-to-one. This intuitive method gives a clear picture of the function's behavior in terms of uniqueness of outputs.

Critical Points

Critical points help identify where a function changes its behavior, such as switching from increasing to decreasing. They are found by setting the derivative equal to zero and solving for \( x \).

In our function, \( h(x) = x^2 - 2x \), the derivative \( h'(x) = 2x - 2 \) is set to zero:\[2x - 2 = 0\]Solving gives \( x = 1 \). This is the critical point where the function changes direction. By analyzing the behavior of the function at and around this point, we can see how the function's slope and direction vary. Since the derivative changes sign at \( x = 1 \) from negative to positive, the critical point indicates a shift from decreasing to increasing, or vice versa.

This change of sign at the critical point directly informs us that the function is not consistently one-to-one, as it does not maintain a single direction (either always increasing or always decreasing), further supporting our analysis.