Question

In Exercises 25–32, graph the function. State the domain and range. $$ h(x)=\frac{8 x+3}{2 x-6} $$

Short answer

The graph of the function h(x) will show a vertical asymptote at \(x=3\). The domain of the function is \(-\infty < x < 3\) and \(3 < x < \infty\), excluding \(x = 3\) where the function is undefined. The range of the function is all real numbers, i.e., \(-\infty < h(x) < \infty\).

Step-by-step solution

Graph the function.

First, the function should be graphed. This can be done by selecting several values for x and evaluating the function at these points, plotting them on a graph. However, the function is undefined when the denominator is zero, i.e., \(x=3\). This results in a vertical asymptote at \(x=3\).

Use the graph to find the domain.

The domain of a function is the set of all real numbers for which the function is defined. In this case, the function h(x) is defined for all real numbers except \(x=3\). Therefore, the domain is \(-\infty < x < 3\) and \(3 < x < \infty\).

Use the graph to find the range.

The range of a function are the 'y' values we can get from the function. For a rational function of this sort, the function approaches negative infinity as \(x\) approaches the asymptote from the left and positive infinity as \(x\) approaches the asymptote from the right. Hence, the range of this function is all real numbers, i.e., \(-\infty < h(x) < \infty\).

Key concepts

Domain and Range

When working with rational functions like \( h(x)=\frac{8x+3}{2x-6} \), understanding the domain and range is crucial. The domain represents all possible x-values for which the function is defined. It's important to remember that a function becomes undefined when its denominator equals zero. For the function \( h(x) \), the expression \( 2x-6 \) causes the function to be undefined when \( 2x-6=0 \).Hence, solving \( 2x-6=0 \) yields \( x=3 \).This means the domain of \( h(x) \) doesn't include 3, but it does include all other real numbers.

• So, the domain is all real numbers except \( x=3 \).
• This can be expressed in interval notation as \( (-\infty, 3) \cup (3, \infty) \).

Similarly, the range of a function is all the possible output values. With rational functions, after removing any restrictions from the vertical asymptotes, the range typically is all real numbers. For \( h(x) \), after considering the graph's behavior, the output can also take any real value. You can express this range as \( (-\infty, \infty) \).

Asymptotes

Asymptotes are lines that a graph approaches but never touches. They are crucial in the context of rational functions. For \( h(x)=\frac{8x+3}{2x-6} \), understanding where the asymptotes lie plays a significant role in graphing.

• Vertical Asymptotes: These occur where the denominator is zero and the function is undefined. For \( h(x) \), we identified a vertical asymptote at \( x=3 \) by solving \( 2x-6=0 \).
• Horizontal Asymptotes: These are determined by looking at the degrees of the polynomial in the numerator and the denominator. Here, both have a degree of 1, suggesting a horizontal asymptote that can be found by evaluating the leading coefficients — \( \frac{8}{2}=4 \). This horizontal line \( y=4 \) is the horizontal asymptote, indicating the behavior of \( h(x) \) for very large or very small \( x \) values.

Understanding asymptotes helps sketch a graph accurately and predict function behavior as it approaches certain values.

Rational Functions

Rational functions are the quotient of two polynomials. In the function \( h(x)=\frac{8x+3}{2x-6} \), the numerator is \( 8x+3 \) and the denominator is \( 2x-6 \). Working with such functions involves knowing how they behave and how they are graphically represented.One key point about rational functions is they often have discontinuities, like holes or asymptotes, caused by their denominators approaching zero. In our case, a discontinuity appears at \( x=3 \) with a vertical asymptote.When graphing, identify these points and plot select values around them for clarity. Notice how near the asymptotes, the function tends to approach infinity or negative infinity, giving insights into rising and falling sharply on the graph.In summary, working with rational functions involves:

• Recognizing discontinuity where the function is undefined.
• Analyzing end behavior governed by horizontal asymptotes.
• Plotting to see the sharp changes caused by vertical asymptotes.

The function behavior and graph intertwine to reveal the full picture of a rational function like \( h(x) \).