Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 3: Problem 17
Using Transformations Use transformations of the graph of \(y=1 / x\) to graph the rational function, and state the domain and range. $$t(x)=\frac{2 x-3}{x-2}$$
Short Answer
Expert verified
Domain: \( x \neq 2 \); Range: \( y \neq 2 \).
Step by step solution
01
Identify Base Function
The given function is \( t(x) = \frac{2x-3}{x-2} \). We start by recognizing that the base function to consider is \( y = \frac{1}{x} \). This is the parent graph for all rational functions of this form.
02
Analyze Horizontal and Vertical Shifts
The function \( \frac{1}{x} \) does not have constants added to its numerator or denominator. In \( t(x) = \frac{2x-3}{x-2} \), the denominator \( x-2 \) indicates a horizontal shift to the right by 2 units. The expression in the numerator hints at further transformations due to changes in the slope or intercept.
03
Determine Asymptotes and Slant Asymptote
To find the vertical asymptote, set \( x-2 = 0 \). Thus, \( x = 2 \) is a vertical asymptote. Now for horizontal asymptotes, since the degrees of numerator and denominator are equal, divide the leading coefficients: \( \frac{2}{1} \) giving \( y=2 \). Therefore, \( y = 2 \) serves as a horizontal asymptote. However, because it's not present in the original, analyze slant by dividing \( 2x-3 \) by \( x-2 \). Dividing, the quotient is \( 2 \), but we'd focus more on graph transformations.
04
Apply Transformations to Graph
The horizontal shift moves the graph of \( y = \frac{1}{x} \) by 2 units right. The vertical shift is determined by the horizontal asymptote, which lies on \( y = 2 \), itself being the line at constant value as \( x \to \infty \). The transformation yields final \( t(x) \) graphically around these new asymptotes.
05
State Domain and Range
The domain excludes values that make the denominator zero. Hence, \( x eq 2 \). Therefore, the domain is all real numbers except \( x = 2 \). The range excludes the horizontal asymptote \( y = 2 \), so it鈥檚 all real numbers except \( y = 2 \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Transformations
Transformations are key to understanding how the graph of a base function, like \( y = \frac{1}{x} \), can change to represent different rational functions. In this exercise, the function \( t(x) = \frac{2x-3}{x-2} \) involves several transformations.
- Horizontal Shift: The standard graph \( y = \frac{1}{x} \) typically has its vertical asymptote at \( x = 0 \). However, in \( t(x) \), the denominator \( x-2 \) indicates a shift 2 units to the right. This means the vertical asymptote now aligns at \( x = 2 \).
- Vertical Shift and Stretch: The numerator \( 2x-3 \) suggests other alterations. Transformations here include a vertical stretch due to the factor of 2, altering how fast the function approaches its horizontal asymptote.
Asymptotes
Asymptotes are lines a graph approaches but never quite reaches. In rational functions, they represent limits of the function's values. For \( t(x) = \frac{2x-3}{x-2} \), identifying asymptotes helps sketch the graph efficiently.
- Vertical Asymptote: This occurs where the function becomes undefined. By solving \( x - 2 = 0 \), we find \( x = 2 \) is the vertical asymptote, indicating values close to this \( x \) will cause the function to spike towards infinity.
- Horizontal Asymptote: Since the degrees of numerator and denominator match, we consider the ratio of their leading coefficients. This gives us \( y = 2 \), meaning the graph will level off and approach \( y = 2 \) as \( x \) moves towards both positive and negative infinity.
Domain and Range
The domain and range of a rational function provide insight into where the function exists and what values it can take.
- Domain: For \( t(x) \), the domain includes all possible \( x \)-values, except where the denominator is zero. Thus, \( x eq 2 \). This ensures that the function has no undefined values due to division by zero.
- Range: The range concerns the \( y \)-values possible from the function. The horizontal asymptote \( y = 2 \) points out that \( t(x) \) will never actually reach this value; hence the range is all real numbers except \( y = 2 \).
