Problem 31 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=\frac{1}{1+x^{2}} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The function given is \( f(x) = \frac{1}{1 + x^2} \). To determine if this function is one-to-one, we need to apply the horizontal line test. A function is one-to-one if no horizontal line intersects the graph of the function more than once.

Step 2: Analyze the Function Behavior

Observe that \( f(x) = \frac{1}{1 + x^2} \) is a rational function. The denominator \( 1 + x^2 \) is always positive for all real \( x \), meaning \( f(x) \) is always positive and less or equal to 1. As \( |x| \) increases, \( x^2 \) increases, making the denominator larger and hence the function value smaller.

Step 3: Sketch the Function Graph

Sketch the graph of \( y = f(x) = \frac{1}{1 + x^2} \). The graph is symmetric with respect to the y-axis because \( f(x) = f(-x) \). The maximum value of the function is 1 at \( x=0 \), and as \( x \) moves away from zero, the value approaches 0 but never reaches 0.

Step 4: Apply the Horizontal Line Test

By examining the graph, draw horizontal lines at different y-values. Particularly, observe lines below 1. A horizontal line can intersect the graph at more than one point (e.g., at \( f(x) = \frac{1}{2} \), \( x = -1 \) and \( x = 1 \) both satisfy this equation). Thus, it fails the horizontal line test.

Key Concepts

Horizontal Line Test

The Horizontal Line Test is a simple graphical method used to determine if a function is one-to-one. A function is considered one-to-one if, and only if, no horizontal line intersects its graph more than once. This means for every output value of the function, there is a unique input value.

To perform the test:

• Sketch the graph of the function.
• Draw several horizontal lines across the graph.
• Check if any line touches the graph at more than one point.

If any horizontal line intersects the graph more than once, the function is not one-to-one. This is because the same output value corresponds to multiple input values, which violates the definition of a one-to-one function. Simple, right? This test is a visual way to ensure that each output is uniquely tied to a specific input.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In this case, our function is given by \[ f(x) = \frac{1}{1+x^2} \].

Here are some key features:

• The denominator, \(1 + x^2\), is never zero for any real number \(x\). This means there are no vertical asymptotes in the graph of this particular function.
• For all real values of \(x\), the denominator \(1+x^2\) is always positive, leading to a positive value for \(f(x)\).
• As the magnitude of \(x\) increases, \(x^2\) becomes larger, increasing the denominator and thus decreasing the value of \(f(x)\).
• The maximum value of this function occurs at \(x=0\), where \(f(x)=1\).

These characteristics help us understand how the function behaves over different intervals, and why it ultimately is not a one-to-one function as per the Horizontal Line Test.

Graph Sketching

Graph sketching is a valuable skill in calculus and pre-calculus because it provides a visual representation of how a function behaves. To sketch the graph of \( y = f(x) = \frac{1}{1+x^2} \), follow these steps:

1. **Identify Symmetries**: Notice that the function is symmetric about the y-axis because \( f(x) = f(-x) \). This tells us that the graph on the left side will mirror the right side.

2. **Determine Maximum and Minimum Values**: The function reaches its maximum value of 1 at \( x = 0 \). As \( x \) moves away from zero in either direction, the value of \( f(x) \) approaches zero but never actually reaches it.

3. **Plot Key Points**: Start by plotting some obvious points such as \((0,1)\), \((-1,0.5)\) and \((1,0.5)\). These help to gain a clearer picture of the curve.

4. **Draw the Curve**: With these points and key characteristics in mind, sketch a smooth, downward-opening curve on either side, showing the behavior as \( x \) goes to positive or negative infinity. The graph will approach but never touch the x-axis as \( x \) increases in absolute value.

The resulting graph is one that gradually curves downwards from a peak at \( (0, 1) \) and asymptotically approaches the x-axis. This visual tool demonstrates why the function does not pass the Horizontal Line Test.