Problem 23 - Elementary and Intermediate Algebra

Question

Determine whether each function is one-to-one. \(f(x)=-x^{2}+3 x\)

Step-by-Step Solution

Answer

The function is not one-to-one.

Step 1: Define a One-to-One Function

A function is one-to-one if every output value corresponds to exactly one input value. This means that for any two different inputs, the outputs must also be different.

Step 2: Analyze the Function

Consider the function: \[ f(x) = -x^2 + 3x \] This is a quadratic function, which typically graphs as a parabola.

Step 3: Check for One-to-One Property Algebraically

A quick test for one-to-one is to see if the function has more than one input for any output value. Let's consider: \[ f(x) = -x^2 + 3x = y \] This can be rearranged to find the x-values that give the same y:\[ x^2 - 3x + y = 0 \]The x-values are solutions to this quadratic equation. If there are two distinct x-values solving this for the same y, it's not one-to-one.

Step 4: Use the Discriminant

The discriminant of the quadratic equation \( ax^2 + bx + c \) given by \( b^2 - 4ac \) determines the number of solutions (x-values). For \( x^2 - 3x + y = 0 \), this becomes:\[ 3^2 - 4(1)(y) = 9 - 4y \]Since \( 9 - 4y \) is positive for many values of y, the discriminant is often positive, indicating two solutions for certain y-values.

Step 5: Conclude from the Analysis

Because there are values of y where the discriminant is positive, indicating more than one x-value can produce the same y, the function \( f(x) = -x^2 + 3x \) is not one-to-one.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that can generally be written in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). The graph of a quadratic function is a parabola. Depending on the sign of \( a \), this parabola will either open upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)). In the case of our function, \( f(x) = -x^2 + 3x \), the leading coefficient \( a \) is negative. This means that the parabola opens downwards. Some key features of quadratic functions include:

• Vertex: This is the highest or lowest point on the graph of the function. For a downward opening parabola, the vertex is the maximum point.
• Axis of Symmetry: A vertical line that passes through the vertex and divides the parabola into two symmetric parts. The equation for the axis of symmetry is \( x = -\frac{b}{2a} \).
• Roots or Zeros: The points where the graph intersects the x-axis, which are the solutions to the equation \( ax^2 + bx + c = 0 \).

Understanding these aspects of quadratic functions helps in analyzing their behaviors, such as whether they are one-to-one, as explored in the exercise.

Discriminant

The discriminant is a crucial concept in determining the nature of the solutions to a quadratic equation. When you have a quadratic equation in the form \( ax^2 + bx + c = 0 \), the discriminant is given by \( b^2 - 4ac \). This value will tell you about the number and nature of the roots of the quadratic equation:

• If the discriminant is positive, the quadratic equation has two distinct real roots. This implies that there are two different x-values that will make the function equal zero.
• If the discriminant equals zero, there is exactly one real root, meaning a repeated real root.
• If the discriminant is negative, the quadratic equation has no real roots, indicating complex or imaginary solutions.

For the function \( f(x) = -x^2 + 3x \), transforming it into the form \( x^2 - 3x + y = 0 \), we get the discriminant as \( 9 - 4y \). In this expression, for many values of \( y \), the discriminant becomes positive, suggesting that there are two different x-values for these y-values, thus reinforcing why the function is not one-to-one.

Function Analysis

Function analysis involves studying the properties and behaviors of a function to understand its characteristics deeply. This may include identifying whether a function is one-to-one, finding its range and domain, and analyzing its graph.To determine if a function is one-to-one, each output value should correspond to exactly one input value. This means for every \( y \), there should be a unique \( x \). One way to test for this property graphically is the horizontal line test. If any horizontal line intersects the graph more than once, the function is not one-to-one.In our exercise, analyzing \( f(x) = -x^2 + 3x \) algebraically showed that the discriminant \( 9 - 4y \) indicates multiple solutions for certain outputs. This implies that for some values of \( y \), there are two different inputs \( x \) that result in the same output, and thus the function is not one-to-one. Understanding function analysis helps in visualizing and conceptualizing how different values affect the output and structure of a algebraic function. This process is vital in various aspects of calculus and algebra, where function behavior dictates many core principles and problem-solving techniques.