Problem 32 - Precalculus : Building Concepts and Connections

Question

State whether each function is one-to-one. $$f(x)=-3 x^{2}+1$$

Step-by-Step Solution

Answer

The function \(f(x) = -3x^2 + 1\) is not one-to-one.

Step 1: Understanding the concept of one-to-one function

A function is said to be one-to-one (or injective) if every element of the range of the function corresponds to exactly one element of the domain. In simpler terms, this means that for every 'y' value in the given function, there exists exactly one 'x' value. This can be tested using the horizontal line test.

Step 2: Applying the horizontal line test

A function is one-to-one if and only if every horizontal line intersects the graph of the function at most once. The function \(f(x) = -3x^2 + 1\) is a quadratic function which forms a parabola opening downwards when graphed. If a horizontal line is drawn anywhere within the range of function values, it will intersect the parabola graph twice, indicating that there are two different 'x' values for a single 'y' value in the function.

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

Since the horizontal line intersects the graph of the function \(f(x) = -3x^2 + 1\) more than once, therefore, the function is not one-to-one.

Key Concepts

Injective Functions

An injective function, also known as a one-to-one function, is characterized by its unique assignment of output values. In simpler terms, every output or 'y' value is linked to only one input or 'x' value.
Imagine a set of individuals each receiving a unique gift. Here, no two individuals share the same gift. Similarly, in a one-to-one function, no two different input values result in the same output. This distinct mapping is crucial in defining injective functions.

• Every 'y' value in the range comes from a unique 'x'.
• If \( f(x_1) = f(x_2) \) then \( x_1 = x_2 \).

Identifying if a function is one-to-one helps in understanding its behavior. Functions that fail to be injective have multiple inputs that give the same output. This lack of uniqueness is what classifies the function as not one-to-one, as demonstrated by the exercise.

Horizontal Line Test

The horizontal line test is a simple yet powerful tool to determine if a function is one-to-one. This visual method involves drawing horizontal lines across the graph of a function.
Picture the graph of the function as a piece of art that needs to pass your criteria of uniqueness. If any of your horizontal lines touch the graph more than once, it fails the test!

• A function passes if each horizontal line intersects the graph at most once.
• Failure to pass indicates that at least one horizontal line crosses the graph more than once.

For the function \( f(x) = -3x^2 + 1 \), this test reveals its non-injective nature. The graph has the shape of a downward-opening parabola. Each horizontal line usually intersects it twice, illustrating multiple inputs yielding the same output.

Quadratic Functions

Quadratic functions form the backbone of many mathematical concepts and real-world applications. Typically expressed as \( ax^2 + bx + c \), these functions generate a parabola when plotted.
In the graph of a quadratic function, the direction depends on the sign of the leading coefficient (\( a \)):

• If \( a \) is positive, the parabola opens upwards.
• If \( a \) is negative, it opens downwards, as in \( f(x) = -3x^2 + 1 \).

This simple change in direction can dictate many properties of the quadratic function, including injectivity. Since parabolas are symmetric, any horizontal line across it can intersect at two points. Especially in the \( f(x) = -3x^2 + 1 \) case, it shows clearly how non-injective quadratic functions can be.