Identify the key features of the graph of the function [tex]f(x)=\log _2(x-1)+2[/tex].
vertical asymptote at [tex]x=1[/tex]
horizontal asymptote at [tex]y=1[/tex]
no [tex]x[/tex]-intercept
no [tex]y[/tex]-intercept
domain of [tex](1, \infty)[/tex]
range of [tex](1, \infty)[/tex]
increasing
Answer (2)
Vertical asymptote at x = 1 .
No horizontal asymptote.
X-intercept at x = 4 5 .
No y -intercept.
Domain of ( 1 , ∞ ) .
Range of ( − ∞ , ∞ ) .
Increasing.
Key features identified are: vertical asymptote at x = 1 , no y-intercept, domain of ( 1 , ∞ ) , and the function is increasing. The other statements were incorrect.
The key features of the graph are: Vertical asymptote at x = 1 , no y-intercept, domain of ( 1 , ∞ ) , increasing
Explanation
Analyzing the Given Information We are given the function f ( x ) = lo g 2 ( x − 1 ) + 2 and a list of its supposed key features. We need to verify each of these features.
Vertical Asymptote The base logarithmic function is y = lo g 2 ( x ) . The given function f ( x ) = lo g 2 ( x − 1 ) + 2 is a transformation of this base function.
The term ( x − 1 ) represents a horizontal shift to the right by 1 unit. This means the vertical asymptote of the base function y = lo g 2 ( x ) , which is x = 0 , is shifted to x = 1 for the given function. So, there is a vertical asymptote at x = 1 .
The term + 2 represents a vertical shift upwards by 2 units. This does not affect the vertical asymptote.
Horizontal Asymptote Logarithmic functions do not have horizontal asymptotes. Therefore, the statement that there is a horizontal asymptote at y = 1 is incorrect.
X-Intercept To find the x-intercept, we set f ( x ) = 0 and solve for x :
lo g 2 ( x − 1 ) + 2 = 0 lo g 2 ( x − 1 ) = − 2 x − 1 = 2 − 2 = 4 1 x = 1 + 4 1 = 4 5 = 1.25 Therefore, there is an x-intercept at x = 4 5 . The statement that there is no x-intercept is incorrect.
Y-Intercept To find the y-intercept, we set x = 0 and solve for f ( 0 ) . However, x = 0 is not in the domain of f ( x ) , since we would be taking the logarithm of a negative number. Therefore, there is no y-intercept. The statement that there is no y-intercept is correct.
Domain The domain of the base function y = lo g 2 ( x ) is ( 0 , ∞ ) . Since the given function has a horizontal shift of 1 unit to the right, the domain of f ( x ) is ( 1 , ∞ ) . The statement that the domain is ( 1 , ∞ ) is correct.
Range The range of the base function y = lo g 2 ( x ) is ( − ∞ , ∞ ) . Vertical shifts do not affect the range, so the range of f ( x ) is ( − ∞ , ∞ ) . The statement that the range is ( 1 , ∞ ) is incorrect.
Increasing Function The base function y = lo g 2 ( x ) is increasing. Horizontal and vertical shifts do not change whether a function is increasing or decreasing. Therefore, the given function is increasing. The statement that the function is increasing is correct.
Final Answer In summary, the correct features are:
Vertical asymptote at x = 1
No y-intercept
Domain of ( 1 , ∞ )
Increasing
The incorrect features are:
Horizontal asymptote at y = 1
No x-intercept
Range of ( 1 , ∞ )
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. Understanding the key features of logarithmic functions, such as asymptotes, intercepts, domain, and range, is crucial for interpreting and applying these models effectively. For example, in seismology, the Richter scale uses logarithms to quantify the magnitude of earthquakes. An increase of one unit on the Richter scale represents a tenfold increase in the amplitude of the seismic waves. By analyzing the logarithmic relationship between earthquake magnitude and wave amplitude, scientists can better understand and communicate the relative sizes of different earthquakes.
The function f ( x ) = lo g 2 ( x − 1 ) + 2 has a vertical asymptote at x = 1 , an x-intercept at x = 4 5 , and no y-intercept. Its domain is ( 1 , ∞ ) and the range is ( − ∞ , ∞ ) . The function is also increasing throughout its domain.
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