Problem 117 - A Graphical Approach to Precalculus with Limits

Question

Although a function may not be one-to-one when defined over its "natural" domain, it may be possible to restrict the domain in such a way that it is one-to-one and the range of the function is unchanged. For example, if we nestrict the domain of the function \(f(x)=x^{2}\) (which is not one-to-one over \((-\infty, \infty)\) to \([0, \infty)\), we obtain a one-to-one function whose range is still \([0, \infty)\) See the figure to the right. Notice that we could choose to restrict the domain of \(f(x)=x^{2}\) to \((-\infty, 0]\) and also obtain the graph of a one-to-one function, except that it would be the left half of the parabola. For each function in Exercises \(117-122\), restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. (GRAPHS CANNOT COPY) $$f(x)=-x^{2}+4$$

Step-by-Step Solution

Answer

Restrict the domain to \((- fty, 0]\) or \([0, fty)\) to make the function one-to-one.

Step 1: Understanding the Function

The function given is \( f(x) = -x^2 + 4 \). This is a downward-facing parabola due to the negative coefficient before the \( x^2 \) term. As such, it is not one-to-one over its natural domain, \((-fty, fty)\). The vertex of this parabola is the point where \( x = 0 \), and the range of the function over its natural domain is \((-fty, 4]\).

Step 2: Identifying the Vertex

The vertex of the function \( f(x) = -x^2 + 4 \) is at the point \( (0, 4) \). This is the highest point on the graph of the parabola, meaning that to find where the function is one-to-one, we need to restrict the domain to either side of this vertex.

Step 3: Choosing a Domain

To make the function one-to-one, we restrict the domain to either the left side \((-fty, 0]\) or the right side \([0, fty)\) of the vertex.- Choosing \((-fty, 0]\) lets us include the left half, starting from \( f(x) = 4 \) and moving downward toward \( -fty \).- Choosing \([0, fty)\) covers the right half of the parabola similarly, but in a positive direction.

Step 4: Verifying Range Preservation

Regardless of whether we choose \((-fty, 0]\) or \([0, fty)\), the range of \( f(x) = -x^2 + 4 \) remains \((-fty, 4] \) since both halves reach the maximum function value of 4 at the vertex.

Step 5: Conclusion

For the function to be one-to-one while keeping the same range, we can choose to restrict the domain to \((-fty, 0]\) or \([0, fty)\). Both options maintain the original range of \((-fty, 4] \).

Key Concepts

Domain Restriction

In mathematics, the domain of a function is the set of input values for which the function is defined. Sometimes, to achieve a desired property in a function, such as making it one-to-one, we need to employ a technique called "domain restriction." This involves limiting the domain so that each input corresponds to exactly one unique output.
For example, consider the function \( f(x) = x^2 \). Naturally, this function is not one-to-one over the domain of all real numbers \(( -\infty, \infty ) \) since, for instance, both \( x = -2 \) and \( x = 2 \) yield the output \( f(x) = 4 \).
By restricting the domain to \([0, \infty)\), we ensure that every x-value leads to a unique y-value, making the function one-to-one. Similarly, for the function \( f(x) = -x^2 + 4 \), by restricting the domain to either side of the vertex point (at \( x = 0 \)), we can make it one-to-one while preserving its range.

Function Range

The range of a function is the set of all possible output values it can produce. Determining the range helps in understanding the extent of values a function can achieve.
In the exercise involving \( f(x) = -x^2 + 4 \), the natural range is \((-\infty, 4] \) due to the parabolic shape of the graph that opens downward. This range is derived from the fact that the vertex of the parabola, at the point \( (0, 4) \), is its maximum.
No matter how we restrict the domain of \( f(x) = -x^2 + 4 \) in efforts to make it one-to-one, its range remains \((-\infty, 4] \). This is because both the left and right sides of the domain reach the maximum function value of 4. Ensuring the range remains unchanged while adjusting the domain is crucial when attempting to maintain certain functional characteristics like one-to-oneness.

Parabola

A parabola is a U-shaped curve that can open upwards or downwards, characterized by the equation \( y = ax^2 + bx + c \). The orientation of this curve is dictated by the coefficient \( a \). A positive \( a \) results in an upward-facing parabola, whereas a negative \( a \) makes it face downward.
The function \( f(x) = -x^2 + 4 \) has a negative coefficient before \( x^2 \), indicating that it faces downward. Its highest point, or vertex, is at \( (0, 4) \). This vertex is pivotal for understanding the symmetry and the maximum or minimum value of the function.
By conceptualizing the parabola, you can better visualize why restricting the domain to one side of the vertex can help make the function one-to-one, as it eliminates the redundancy of y-values corresponding to multiple x-values.

Function Graphing

Graphing a function is an essential skill in understanding its behavior visually. Graphs help display crucial aspects such as intercepts, slopes, and symmetries. The function \( f(x) = -x^2 + 4 \) forms a downward-facing parabola.
Visualizing the graph can help identify key points, such as where the function changes direction (the vertex) and where it crosses the y-axis (the y-intercept at \( f(x) = 4 \)).

• To determine where a function can be one-to-one, look at the symmetry of the graph and decide how the domain needs to be restricted around the vertex.
• Using graphing tools or plotting by hand can provide insights into potential domain restrictions and confirm the range.
• Observing graph points also aids in identifying unnecessary overlaps in mapping x-values to y-values.

Graphing allows the combination of these observations, creating a comprehensive understanding of the function's dynamics.