Problem 31 - Precalculus : Building Concepts and Connections

Question

State whether each function is one-to-one. $$f(x)=2 x^{2}-3$$

Step-by-Step Solution

Answer

No, the function \( f(x) = 2x^2 - 3 \) is not a one-to-one function.

Step 1: Graph the function

First, we will graph the function \( f(x) = 2x^2 - 3 \). This function is a parabola that opens upwards and is shifted 3 units down.

Step 2: Apply the horizontal line test

Now, we will apply the horizontal line test to the graph of the function. This test involves imagining or drawing a horizontal line that crosses the graph. If this horizontal line intersects the graph at more than one point, then the function is not one-to-one.

Step 3: Conclude if the function is one-to-one

By applying the horizontal line test to the function, we can observe that horizontal lines cross the graph at more than one point. Therefore, the function \( f(x) = 2x^2 - 3 \) is not a one-to-one function.

Key Concepts

Horizontal Line Test

The horizontal line test is a simple graphical method to determine if a function is one-to-one. A one-to-one function means that each output value is paired with exactly one input value. To perform this test, you imagine or draw horizontal lines across the graph of a function. If any horizontal line intersects the graph at more than one point, the function is not one-to-one. This is because multiple x-values yield the same y-value.

• If a horizontal line touches the graph at just one point, the function passes the test and is one-to-one.
• If a horizontal line intersects the graph at more than one point, it fails the test and is not one-to-one.

For the function \( f(x) = 2x^2 - 3 \), any horizontal line will intersect the parabola in two places. Therefore, it is not a one-to-one function.

Graphing Functions

Graphing functions is an essential skill for understanding and visualizing mathematical relationships. The function \( f(x) = 2x^2 - 3 \) is a great example to demonstrate this. It is expressed in standard quadratic form, \( f(x) = ax^2 + bx + c \). Here's how to graph it:

• This function is a parabola that opens upwards, because the coefficient of \( x^2 \) (which is 2 in this case) is positive.
• To graph it, you start by plotting the vertex, which for a parabola in this form, is located at \( (0, -3) \), as it is the point of minimum value.
• The parabola is symmetric around its vertex line, which in this case is the y-axis.

You can create a quick sketch by plotting additional points: calculate values for \( x \) such as -1, 1, -2, and 2, and find their corresponding \( y \)-values to get a clearer shape of the curve.

Parabola

A parabola is a U-shaped curve that comes from quadratic functions. Understanding its properties can be helpful:

• For the equation \( f(x) = ax^2 + bx + c \), the parabola opens upwards if \( a > 0 \), and downwards if \( a < 0 \).
• The vertex of a parabola, \( f(x) = 2x^2 - 3 \), is a critical point that represents its maximum or minimum value. Here, the vertex is \( (0, -3) \).
• Parabolas are symmetrical. This means that the left and right sides of the vertex have the same shape.

In this specific problem, the parabola opens upwards, confirming the function is not one-to-one, as shown by its graph intersecting multiple points when applying the horizontal line test.