Problem 7
Question
Use a calculator and the Horizontal Line Test to determine whether or not the function \(f\) is one-to-one. $$f(x)=.1 x^{3}-.1 x^{2}-.005 x+1$$
Step-by-Step Solution
Verified Answer
Answer: (Student's response based on the graph and analysis of the horizontal line test)
1Step 1: Graph the function
To determine whether the function is one-to-one, start by graphing \(f(x) = .1x^3 - .1x^2 - .005x + 1\). You can use a graphing calculator or an online graph plotter like Desmos to generate the graph.
2Step 2: Perform the Horizontal Line Test
Observe the graph and perform the horizontal line test. The horizontal line test states that if a horizontal line can be drawn in such a way that it intersects the curve more than once, the function is not one-to-one.
3Step 3: Analyze the results
If the graph of the function never intersects a horizontal line more than once, then it passes the horizontal line test, and the function is one-to-one. On the other hand, if there is any horizontal line that intersects the graph more than once, the function is not one-to-one.
Based on the graph and the results of the horizontal line test, determine if the function is one-to-one. This should give you the final answer to the exercise.
Key Concepts
Horizontal Line TestGraphing Polynomial FunctionsUsing Graphing Calculators
Horizontal Line Test
Understanding the Horizontal Line Test is pivotal when studying one-to-one functions. By definition, a function is one-to-one if each input corresponds to exactly one output and each output corresponds to exactly one input. To visually determine if a function has this property, we employ the Horizontal Line Test.
Imagine taking a horizontal line and sweeping it up and down across the graph of the function. If at any point the line touches the graph in more than one place simultaneously, it indicates that the function is not one-to-one—meaning a single output is shared by multiple inputs, which violates the definition of a one-to-one function. Conversely, if every horizontal line drawn only ever touches the graph at one point, congrats: You're dealing with a one-to-one function! This test is not just a theoretical concept; it has practical uses. For example, only one-to-one functions have inverses that are also functions.
Imagine taking a horizontal line and sweeping it up and down across the graph of the function. If at any point the line touches the graph in more than one place simultaneously, it indicates that the function is not one-to-one—meaning a single output is shared by multiple inputs, which violates the definition of a one-to-one function. Conversely, if every horizontal line drawn only ever touches the graph at one point, congrats: You're dealing with a one-to-one function! This test is not just a theoretical concept; it has practical uses. For example, only one-to-one functions have inverses that are also functions.
Graphing Polynomial Functions
When tackling polynomial functions, such as the one mentioned in the exercise, the art of graphing becomes both an art and a science. Polynomial functions can have curves (known as parabolas), lines, and even cusp or turning points. The function provided, which is a cubic polynomial function, will typically resemble a wave or an 'S' shape on a graph. This type of function can cross the x-axis multiple times, offering several real number solutions to the equation \( f(x) = 0 \).
The important features to look out for when sketching the graph of a polynomial function include the x-intercepts (where the function crosses the x-axis), y-intercept (where the function crosses the y-axis), and the end behavior (how the function behaves as \( x \) approaches infinity or negative infinity). By determining the shape and where the function intersects the axes, you can build a clear picture of the function's graph, which is essential for applying tests like the Horizontal Line Test.
The important features to look out for when sketching the graph of a polynomial function include the x-intercepts (where the function crosses the x-axis), y-intercept (where the function crosses the y-axis), and the end behavior (how the function behaves as \( x \) approaches infinity or negative infinity). By determining the shape and where the function intersects the axes, you can build a clear picture of the function's graph, which is essential for applying tests like the Horizontal Line Test.
Using Graphing Calculators
Graphing calculators are a fantastic tool for visualizing functions and applying concepts like the Horizontal Line Test effectively. By inputting the function's formula, in this case, \( f(x) = 0.1x^3 - 0.1x^2 - 0.005x + 1 \), you can see the graph on the calculator's screen. Devices like the TI-84 or software like Desmos can handle these tasks with ease.
- Start by entering the function into the graphing calculator.
- Adjust the viewing window to ensure you can see the relevant parts of the graph, including where it crosses the axes.
- Use the trace or examine function to see how the graph behaves in detail at various points
Other exercises in this chapter
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