Problem 162 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the function. $$f(x)=\left\\{\begin{array}{l}x-9, x \leq 1 \\\x^{2}+1, x>1\end{array}\right.$$

Step-by-Step Solution

Answer

The graph should consist of a line segment for \(x \leq 1\) that is obtained from \(f(x) = x - 9\), and a parabola for \(x > 1\) that is obtained from \(f(x) = x^{2} + 1\), with a hole at \(x = 1\) on the parabola.

Step 1: Identify Ranges

First, identify the ranges for \(x\) in the piecewise function. We have two parts here: \(x-9\) applies to \(x \leq 1\) and \(x^{2}+1\) applies to \(x > 1\). These two parts will be graphed separately.

Step 2: Graph First Part

Next, we graph the function \(f(x) = x - 9\) for \(x \leq 1\). This is a simple linear function with a slope of 1 and a y-intercept of -9. We plot this line for all \(x\) values that are less than or equal to 1.

Step 3: Graph Second Part

Then, we graph the function \(f(x) = x^{2} + 1\) for \(x > 1\). This is a quadratic function and its graph is a parabola opening upwards with its vertex at (0,1). We plot this parabola for all \(x\) values that are greater than 1. As \(x = 1\) is not included in this range, we indicate this on the graph with an open circle or a hole at the point corresponding to \(x = 1\).

Step 4: Combine Graphs

Finally, we merge the two graphs. Each part of this graph corresponds to the appropriate function based on the value of \(x\). Thus our final graph consists of a straight line for \(x \leq 1\) and a parabola for \(x > 1\), with a hole at \(x = 1\) on the parabola.

Key Concepts

Graphing Functions

Graphing a function means drawing its visual representation, allowing you to see the behavior of the function. In the case of piecewise functions such as \(f(x)=\begin{cases} x-9, & x \leq 1 \ x^{2}+1, & x > 1 \end{cases}\), the graph is created by combining individual graphs for each specified section of the piecewise function. By understanding the intervals—in this case, \(x \leq 1\) and \(x > 1\)—we can plot each function over its respective domain.

To accurately graph this function, it is critical to:

• Identify the different formulas or expressions based on the value of \(x\).
• Graph each part separately within its specified range.
• Recognize any discontinuities, such as holes for points not included in specific intervals.

Linear Functions

A linear function represents a straight line when graphed. It can be described by the formula \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the function \(f(x) = x - 9\), which applies for \(x \leq 1\), the slope is 1 and the y-intercept is -9. This means:

• The line rises one unit upward for every one unit it moves to the right.
• At \(x = 0\), the function value, \(f(x) = 0 - 9 = -9\), crosses the y-axis.

Due to the linearity, the graph is straightforward. Plotting this on the coordinate plane involves marking points for various \(x\) values (such as \(x = -1, 0, 1\)), drawing a line through those points, and extending it towards \(x = -\infty\) within the restriction \(x \leq 1\).

Quadratic Functions

Quadratic functions form a parabolic shape on a graph, typically described by \(f(x) = ax^2 + bx + c\). In our piecewise function, the quadratic part is \(f(x) = x^2 + 1\) for \(x > 1\). This particular function has:

• A leading coefficient \(a = 1\), indicating the parabola opens upwards.
• A vertex that would be at \((0, 1)\) if graphed entirely.

Since the domain is restricted to \(x > 1\), we do not include \((1, 2)\), but instead mark it with an open circle to signify it's not on the graph. As \(x\) increases, the function values get larger, causing the curve to arch upwards as it moves to the right.

Function Analysis

Analyzing a piecewise function involves understanding both its individual parts and its overall behavior. Key considerations include:

• Domain: Each part has specific domains, with \(x - 9\) applying to \(x \leq 1\) and \(x^2 + 1\) to \(x > 1\).
• Continuity: At \(x = 1\), there's a transition from linear to quadratic, creating a point of discontinuity for the parabola.
• Endpoints and Range: While the end of the linear part at \(x = 1\) is filled, the start of the quadratic at \(x = 1\) is not, which can influence how we interpret the function's range and endpoints.

Overall, a piecewise function such as this demands careful consideration of how its components juxtapose to inform its graph and application, leading to a visual piece that requires detailed attention and precision.