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Magazine Chapter 1: Problem 4
Sketch a graph of a function that is one-to-one on the intervals \((-\infty,-2]\) and \([-2, \infty)\) but is not one-to-one on \((-\infty, \infty)\).
Short Answer
Expert verified
Answer: Yes
Step by step solution
01
Identify a possible function that meets the criteria
A possible function that meets the criteria is a piecewise function defined by two quadratic functions. For the interval \((-\infty,-2]\), let's use the downward-opening parabola \(f(x) = (x+2)^2\). For the interval \([-2, \infty)\), the function can be an upward-opening parabola \(f(x) = (x-2)^2\). This function will be one-to-one on both intervals, but not overall, due to the parabolas repeating their output values.
02
Define the piecewise function
Let's define the piecewise function:
$$
f(x) = \begin{cases}
(x+2)^2 &\text{for } x \in (-\infty,-2]\\
(x-2)^2 &\text{for } x \in [-2, \infty)
\end{cases}
$$
03
Sketch the graph of the function
We will now create a sketch of the function. To do this, we will first sketch the individual parabolas and then combine them to form our desired function:
1. For the interval \((-\infty,-2]\):
- Vertex at \((-2, 0)\)
- Opens downward
2. For the interval \([-2, \infty)\):
- Vertex at \((2, 0)\)
- Opens upward
Note that the function will be continuous at the value \(x=-2\) as both halves of the function are equal to zero.
Here is the graph of the function:
[Graph of the piecewise function with the left part being a downward-opening parabola with vertex at (-2,0) and the right part being an upward-opening parabola with vertex at (2,0).]
The graph of this piecewise function meets the given criteria, and we have successfully sketched a function that is one-to-one on the intervals \((-\infty,-2]\) and \([-2, \infty)\) but is not one-to-one on \((-\infty, \infty)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
One-to-One Functions
In mathematics, a one-to-one function, also known as an injective function, has a distinct characteristic: each element of the function's domain maps to a unique element in the range. In simpler terms, no two different inputs (or x-values) in the domain produce the same output (or y-value).
This is important for determining if an inverse of the function exists over a specific interval. For a function to have an inverse that is also a function, the original function must be one-to-one.
This is important for determining if an inverse of the function exists over a specific interval. For a function to have an inverse that is also a function, the original function must be one-to-one.
- If you draw a horizontal line anywhere on the graph of the function and it crosses the curve more than once, the function is not one-to-one.
- However, when restricting the domain, as we did in the piecewise function, each segment can ensure a unique mapping, thus making each segment one-to-one.
Quadratic Functions
Quadratic functions are a type of polynomial function with the degree of two, typically written in the form \(f(x) = ax^2 + bx + c\). The graph of a quadratic function is a parabola, which can open upwards or downwards based on the sign of the coefficient \(a\).
In our piecewise example:
In our piecewise example:
- The function on the interval \((-\infty,-2]\) is \((x+2)^2\), a quadratic that opens downward.
- On the interval \([-2, \infty)\), \((x-2)^2\) represents another quadratic, this time opening upwards.
- For \((x+2)^2\), the vertex is at \((-2, 0)\).
- For \((x-2)^2\), the vertex is at \((2, 0)\).
Graph Sketching
Graph sketching involves drawing a rough representation of a function's overall shape and behavior. It's a valuable skill for visualizing the concepts of functions. When sketching graphs, especially for piecewise functions:
It's crucial to ensure continuity, especially when segments meet, like at \(x = -2\) in our function. Proper graph sketching helps clarify the relationship between the mathematical expressions and their graphical interpretations, a powerful tool in understanding complex math concepts.
- Identify the individual segments and their equations.
- Determine key points such as vertices, intercepts, and any restrictions on the domain.
- Consider the direction each segment opens (upwards or downwards for quadratics).
It's crucial to ensure continuity, especially when segments meet, like at \(x = -2\) in our function. Proper graph sketching helps clarify the relationship between the mathematical expressions and their graphical interpretations, a powerful tool in understanding complex math concepts.
