Problem 27
Question
In Exercises 25–32, graph the function. State the domain and range. $$ y=\frac{x+6}{4 x-8} $$
Step-by-Step Solution
Verified Answer
The vertical asymptote is \( x = 2 \). The domain is \( x \neq 2 \), and the range is all real numbers. The graph of the function is a hyperbola with vertical asymptote at \( x=2 \).
1Step 1: Find the Vertical Asymptote
Solve \( 4x-8=0 \) to find the vertical asymptote, which is \( x=2 \). This is because rational functions have vertical asymptote where the denominator is zero.
2Step 2: Determine the Domain
The domain is all real number except the x value of the vertical asymptote. So, the domain is \( x \neq 2 \). The x value 2 is excluded because it leads to division by zero which is undefined in mathematics.
3Step 3: Determine the Range
Looking at our function it may be seen that the outputs could be any real number, so the range is all real numbers.
4Step 4: Graph the Function
To graph this function, plot the vertical asymptote at x=2. Then plot a few points on either side of the asymptote. As x approaches 2 from the left or right, the y-values go to negative or positive infinity. This graph is a hyperbola.
Key Concepts
Vertical AsymptoteDomain and RangeGraphing Hyperbolas
Vertical Asymptote
A vertical asymptote is a vertical line on a graph where the values of a function approach infinity as the input gets closer to a certain value. In rational functions like \( y=\frac{x+6}{4x-8} \), vertical asymptotes occur where the denominator of the fraction equals zero. These points are crucial as they help us understand the behavior of the graph near certain x-values.
To find a vertical asymptote, solve for the x-value that makes the denominator zero. In this function, setting \( 4x-8=0 \) gives \( x=2 \). This means as x approaches 2, the function's output goes towards positive or negative infinity. Recognizing this point on the graph involves:
To find a vertical asymptote, solve for the x-value that makes the denominator zero. In this function, setting \( 4x-8=0 \) gives \( x=2 \). This means as x approaches 2, the function's output goes towards positive or negative infinity. Recognizing this point on the graph involves:
- Setting the denominator equal to zero
- Solving for x
- Drawing a dashed line on the graph at this x-value
Domain and Range
The domain and range are fundamental concepts in understanding the scope and scale of functions.
The **domain** of a function refers to all the possible input values (x-values) that can be plugged into the function without causing any errors such as division by zero. For the function \( y=\frac{x+6}{4x-8} \), the domain is all real numbers except \( x=2 \), expressed as \( x eq 2 \). This exclusion is because having \( x=2 \) makes the denominator zero, leading to an undefined scenario.
The **range** is all possible output values (y-values) a function can produce. In our rational function example, although calculating range can sometimes be complex, in this context, observe that there are no restrictions on y-values based directly on the function's structure. Thus, the range is all real numbers, meaning any real number could potentially be a result of y.
The **domain** of a function refers to all the possible input values (x-values) that can be plugged into the function without causing any errors such as division by zero. For the function \( y=\frac{x+6}{4x-8} \), the domain is all real numbers except \( x=2 \), expressed as \( x eq 2 \). This exclusion is because having \( x=2 \) makes the denominator zero, leading to an undefined scenario.
The **range** is all possible output values (y-values) a function can produce. In our rational function example, although calculating range can sometimes be complex, in this context, observe that there are no restrictions on y-values based directly on the function's structure. Thus, the range is all real numbers, meaning any real number could potentially be a result of y.
Graphing Hyperbolas
Graphing hyperbolas, particularly when they arise from rational functions, involves understanding how these curves behave near asymptotes.
For the function \( y=\frac{x+6}{4x-8} \), graphing starts by marking the vertical asymptote at \( x=2 \). Next, select several x-values on both sides of this asymptote to compute corresponding y-values. Plot these points on the coordinate plane.
Observe that the graph forms two distinct curves that mirror each other with the vertical asymptote standing between them. As x moves closer to 2 from either side, the y-values will shoot up to infinity or dive down to negative infinity, indicating the function’s rapid increase or decrease.
When graphing:
For the function \( y=\frac{x+6}{4x-8} \), graphing starts by marking the vertical asymptote at \( x=2 \). Next, select several x-values on both sides of this asymptote to compute corresponding y-values. Plot these points on the coordinate plane.
Observe that the graph forms two distinct curves that mirror each other with the vertical asymptote standing between them. As x moves closer to 2 from either side, the y-values will shoot up to infinity or dive down to negative infinity, indicating the function’s rapid increase or decrease.
When graphing:
- Note the approach of curves towards asymptotes without touching them
- Ensure the shape resembling hyperbola branches
- Assess symmetry if any, about vertical or horizontal axes
Other exercises in this chapter
Problem 26
Find the sum or difference. \(\frac{x+3}{x^2-25}-\frac{x-1}{x-5}+\frac{3}{x+3}\)
View solution Problem 26
Solve the equation by using the LCD. Check your solution(s). $$\frac{10}{x^2-2 x}+\frac{4}{x}=\frac{5}{x-2}$$
View solution Problem 27
Solve the equation by using the LCD. Check your solution(s). $$\frac{x+1}{x+6}+\frac{1}{x}=\frac{2 x+1}{x+6}$$
View solution Problem 27
Find the quotient. $$ \frac{32 x^3 y}{y^8} \div \frac{y^7}{8 x^4} $$
View solution