Problem 21 - A Graphical Approach to College Algebra

Question

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}}\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=\frac{1}{x}+1$$

Step-by-Step Solution

Answer

Shift the graph of \( y = \frac{1}{x} \) up by 1 unit. Domain: \( x \neq 0 \); Range: \( y \neq 1 \).

Step 1: Start with Basic Graph

First, identify the basic graph that we will use as our starting point. The function given is \( f(x) = \frac{1}{x} + 1 \), which means we will start with the graph of \( y = \frac{1}{x} \). This is a hyperbolic curve that has two branches, one in the first quadrant (where \( x > 0 \)) and one in the third quadrant (where \( x < 0 \)). The graph has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \).

Step 2: Understand Transformation

The function \( f(x) = \frac{1}{x} + 1 \) involves a vertical shift of the graph of \( y = \frac{1}{x} \). The '+1' in the function indicates that each point on the graph of \( y = \frac{1}{x} \) is shifted up by 1 unit. This changes the position of the horizontal asymptote from \( y = 0 \) to \( y = 1 \). The vertical asymptote remains unchanged at \( x = 0 \).

Step 3: Sketch the Transformed Graph

To draw the graph by hand, start with the vertical asymptote at \( x = 0 \) as a dashed line. Then, draw the two branches of the hyperbola that resemble the shape of \( y = \frac{1}{x} \), but shifted upward so that they approach the horizontal line \( y = 1 \) as the new asymptote. The first branch should be in the first quadrant approaching \( x = 0 \) from the right, and the second branch should be in the third quadrant approaching \( x = 0 \) from the left.

Step 4: Graph Using a Calculator

Use a graphing calculator to plot the function \( f(x) = \frac{1}{x} + 1 \). Enter the function into the calculator, and observe the accurate depiction of the graph. Ensure your plot reflects the vertical shift and asymptotes described earlier.

Step 5: Identify the Domain and Range

Determine the domain and range of the function. The domain of \( f(x) = \frac{1}{x} + 1 \) is all real numbers except \( x = 0 \), as the function is undefined there. The range, as shifted from \( y = \frac{1}{x} \), is all real numbers except \( y = 1 \). The function never actually reaches \( y = 1 \) due to the horizontal asymptote.

Key Concepts

Graphing Functions

Graphing functions is a way to visualize mathematical relationships. It provides a picture of how a function behaves for different input values of the variable. In this context, we started with the graph of the basic function, which is hyperbola-shaped:

• The function curve is defined by the equation \(y = \frac{1}{x}\).
• Two branches of the hyperbola, one in the first quadrant and the other in the third quadrant, appear due to the positive and negative values of \(x\).
• A key feature is the vertical asymptote at \(x = 0\), where the graph approaches but never touches. This line indicates where the function is undefined.
• Another notable aspect is the horizontal asymptote at \(y = 0\), representing the value that \(y\) approaches as \(x\) becomes extremely large or small.

By using graphing calculators or manually sketching the graph, students can better observe these characteristics and how they change when functions modify through transformations.

Vertical Shift

Vertical shifts in graphing a function involve moving the entire graph up or down without changing its shape. For the function \(f(x) = \frac{1}{x} + 1\):

• The \(+1\) signifies a shift upwards by 1 unit.
• This transformation affects the horizontal asymptote, moving it from \(y = 0\) to \(y = 1\).
• It means every point on the graph of \(y = \frac{1}{x}\) is elevated by one unit.
• Importantly, the vertical asymptote remains unchanged, as it is only influenced by horizontal movements.

Graphically, vertical shifts are simple to recognize since they keep the overall graph orientation intact while lifting it higher or moving it lower. This particular shift provides a clear physical interpretation of adding a constant to a function, visually clarifiable with both sketches and graphing tools.

Domain and Range

The domain and range are fundamental concepts when analyzing functions. **Domain** refers to the set of all possible input values (\(x\)-values) that a function can accept, while **range** is the set of all possible output values (\(y\)-values) the function can produce. In the case of \( f(x) = \frac{1}{x} + 1 \):

• **Domain:** All real numbers except \(x = 0\). The function is undefined at \(x = 0\), as dividing by zero is impossible.
• **Range:** All real numbers except \(y = 1\). Asymptotic behavior ensures that \(y = 1\) is never reached, due to the transformation bringing the entire graph 1 unit up.

Understanding these sets helps in knowing the limits within which the function operates. The insights into domain and range often assist in many directions from resolving complex equations to evidencing the constraints in real-world modeling.