Problem 13 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

In Exercises 11–18, graph the function. State the domain and range. $$ h(x)=\frac{6}{x-1} $$

Step-by-Step Solution

Answer

The graph of \(h(x) = \frac{6}{x-1}\) is a hyperbola with domain as \((-∞, 1) ∪ (1, ∞)\) and range as \((-∞, ∞)\) with vertical asymptote at \(x = 1\) and horizontal asymptote at \(y = 0\).

Step 1: Find the Domain

The domain of \(h(x)\) is all the real numbers except the value that makes the denominator equal to zero since division by zero is undefined. So, we set the denominator equal to zero and solve for \(x\): \(x - 1 = 0\). From that, we find that \(x = 1\). Hence, the domain of \(h(x)\) is \(x\) such that \(x ≠ 1\). In interval notation, this is written as \((-∞, 1) ∪ (1, ∞)\).

Step 2: Find the Range

The range of \(h(x) = \frac{6}{x-1}\) is all real numbers. This is because the numerator is a non-zero constant and the denominator can take any value except zero. Therefore, the function \(h(x)\) can take any real value. The range is represented as \((-∞, ∞)\).

Step 3: Identify the Asymptotes

For the given function, the vertical asymptote is found by setting the denominator equal to zero, giving us \(x = 1\). The horizontal asymptote for the function \(h(x) = \frac{6}{x-1}\) as \(x-h\) goes to ±∞ is \(y = 0\). These asymptotes will be useful when drawing the graph of the function.

Step 4: Draw the Graph

The graph of \(h(x) = \frac{6}{x-1}\) is a hyperbola with a vertical asymptote at \(x = 1\) and a horizontal asymptote at \(y = 0\). The hyperbola approaches these asymptotes as \(x\) moves away from \(x = 1\) but never crosses them.

Key Concepts

Domain and Range

When working with rational functions like \( h(x) = \frac{6}{x-1} \), it's important to determine the domain and range. The domain consists of all possible values that \( x \) can take, without making the function undefined. In this case, the function becomes undefined if the denominator is zero. Solving \( x - 1 = 0 \), we find that \( x = 1 \). So, \( x \) cannot be equal to 1. In interval notation, this domain is expressed as \((-\infty, 1) \cup (1, \infty)\).

Now, let's identify the range. The range of a function is the set of values the function can produce. Here, the numerator is a constant, while the denominator can take any real number except zero. This means \( h(x) \) can take any real value, therefore, the range is \((-\infty, \infty)\).

Understanding the domain and range is crucial. It helps us predict where the function is and isn't defined and what values the function can output.

Vertical and Horizontal Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually touches. Identifying these in rational functions like \( h(x) = \frac{6}{x-1} \) can significantly aid in graphing.

• Vertical Asymptotes: These occur where the function goes to infinity. For \( h(x) \), the vertical asymptote is found by setting the denominator to zero, \( x - 1 = 0 \), hence \( x = 1 \). This means as \( x \) approaches 1, \( h(x) \) becomes infinitely large or small.
• Horizontal Asymptotes: These occur when \( x \) tends to infinity, indicating the value the function settles at. For \( h(x) = \frac{6}{x-1} \), as \( x \) trends toward plus or minus infinity, \( y \) approaches 0. The horizontal asymptote, therefore, is \( y = 0 \).

Recognizing where these asymptotes are helps us understand the behavior of the graph near critical points.

Graphing Hyperbolas

Graphing the function \( h(x) = \frac{6}{x-1} \), you'll notice it forms a hyperbola. Hyperbolas are curves formed by the graph of rational functions and have distinct asymptotes.
The steps to graph them include:

- **Plot the Asymptotes:** Begin by drawing the vertical and horizontal asymptotes found earlier. These lines guide how the graph is shaped. For \( h(x) \), draw a vertical line at \( x = 1 \) and a horizontal line at \( y = 0 \).

- **Sketch the Curve:** Visualize the function getting closer to these lines but never touching them. The graph appears in two separate pieces, with one part in quadrants II and IV, and the other in quadrants I and III.

Hyperbolas appear symmetric relative to their center, which in this function isn’t explicitly visible but hinted by the asymptotic lines. Understanding these shapes can help when sketching and interpreting rational function graphs.