Question

In Exercises 25–32, graph the function. State the domain and range. $$ f(x)=\frac{-5 x+2}{4 x+5} $$

Short answer

The graph of the function \(f(x) = \frac{-5x + 2}{4x+5}\) will be a hyperbolic type graph with a gap at \(x = -\frac{5}{4}\). The domain is \((- \infty, -\frac{5}{4}) \cup (-\frac{5}{4}, \infty)\) and the range is also \((- \infty, -\frac{5}{4}) \cup (-\frac{5}{4}, \infty)\).

Step-by-step solution

Determine the Domain

For a rational function like this one, the value of x cannot make the denominator (4x + 5 in this case) equal to zero, since division by zero is undefined. Therefore, solve the equation \(4x + 5 = 0\) to find the values that should be excluded from the domain. Solving this gives us \(x = -\frac{5}{4}\). Therefore, the domain will be all real numbers except when \(x = -\frac{5}{4}\). In interval notation, this is \((- \infty, -\frac{5}{4}) \cup (-\frac{5}{4}, \infty)\).

Graph the function

To graph the function, first identify a series of x-values to substitute into the function to get the corresponding f(x)-values. For example, choose values for x before and after -5/4, (such as, -2, -1 and 0) to graph the function since the function will not be defined at x = -5/4. Plot these points on the graph and connect them smoothly to create the curve.

Determine the Range

The range of a graphed function consists of all possible output values or y-values. Looking to the graphical representation, it becomes clear that the function covers all y-values except for a certain value which is the horizontal asymptote. To find this, look at what happens as x tends to positive or negative infinity. Both terms of the equation will produce a result of \(-\frac{5}{4}\), and as x gets larger the constant terms have less effect and the ratio of the leading terms will have a greater effect. Hence, the function tends to \(-\frac{5}{4}\) as x tends to positive or negative infinity. This means the range of the function is all real numbers except for when \(f(x) = -\frac{5}{4}\). In interval notation, this would be \((- \infty, -\frac{5}{4}) \cup (-\frac{5}{4}, \infty)\).

Key concepts

Domain of a Function

When we talk about the domain of a function, we're referring to the set of all possible input values (usually 'x' values) that the function can accept without causing any mathematical issues, like division by zero or taking the square root of a negative number in the context of real numbers.

For rational functions, which are ratios of two polynomials, the domain is all real numbers except where the denominator is zero, since division by zero is undefined. In the case of the function \( f(x)=\frac{-5x+2}{4x+5} \), the denominator becomes zero at \( x=-\frac{5}{4} \). Thus, the domain excludes this value. Using interval notation, which is a way of describing sets of numbers through intervals, we write the domain as \( (-\infty, -\frac{5}{4}) \cup (-\frac{5}{4}, \infty) \), signifying all real numbers except \( -\frac{5}{4} \).

Range of a Function

The range of a function, on the other hand, is all the possible outputs (typically 'y' values) that the function can produce. Determining the range is about understanding what values the 'y', or the output of the function, can take.

For the same function \( f(x)=\frac{-5x+2}{4x+5} \), we observe the range by examining the graph or by considering the behavior of the function as the inputs approach large positive or negative values. In this case, the range does not include \( y=-\frac{5}{4} \), which happens to be a horizontal asymptote. So, similarly to the domain, we express the range in interval notation as \( (-\infty, -\frac{5}{4}) \cup (-\frac{5}{4}, \infty) \).

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. These can be vertical, horizontal, or oblique (slant).

Vertical asymptotes occur at values that are not in the domain of the function and are typically where the function goes towards infinity. In our example, the vertical asymptote is the line \( x=-\frac{5}{4} \) because the function is undefined there and the graph will shoot off towards infinity as it approaches this line from either direction.

Horizontal asymptotes are a bit different. They show the value that the output (y) of a function approaches as the input (x) goes to positive or negative infinity. In this example, \( y=-\frac{5}{4} \) is a horizontal asymptote, meaning the graph of the function will get closer and closer to the line \( y=-\frac{5}{4} \) as x moves towards \( \pm\infty \).

Interval Notation

Interval notation is a system of writing the domain and range of functions that describe intervals of values concisely. This type of notation is particularly useful when dealing with continuous sets of numbers.

It typically includes a combination of parentheses '()' and square brackets '[]', where the former is used to denote that an endpoint is not included ('open interval') and the latter implies an endpoint is included ('closed interval'). For example, the notation \((a, b)\) indicates all numbers greater than 'a' and less than 'b', not including 'a' and 'b' themselves.

In the function \( f(x)=\frac{-5x+2}{4x+5} \), interval notation was used to represent its domain and range which excluded the values where vertical and horizontal asymptotes occur, respectively. The domain and range were denoted as \((-\infty, -\frac{5}{4}) \cup (-\frac{5}{4}, \infty)\), indicating all real numbers except \(-\frac{5}{4}\).