The domain of the function f(x)=1−log5(x−2) is (2,∞) .
Explanation of Solution
Given information:
The function is f(x)=1−log5(x−2) .
Consider the function f(x)=1−log5(x−2) .
The given function is a logarithmic function.
The logarithmic function is defined only for positive real numbers.
The domain of f consists of all x for which x−2>0 , or equivalently, x>2 .
The domain of f is {x|x>2} , or equivalently, (2,∞) in interval notation.
Thus, the domain of the function f(x)=1−log5(x−2) is (2,∞) .
(b)
To determine
To graph: The function f(x)=1−log5(x−2) .
(b)
Expert Solution
Explanation of Solution
Given information:
The function is f(x)=1−log5(x−2) .
Graph:
Use the steps below to graph the function using graphing calculator:
Step I: Press the ON key.
Step II: Now, press [Y=]. Input the right hand side of the function y=1−log5(x−2) in Y1 .
Step III: Then hit [Graph] key to view the graph.
The graph of the function f(x)=1−log5(x−2) is as follows:
Interpretation:
By looking at the graph, it is observed that as x→2 , y→∞ . As x value increases from x=2 , y also increases. The above graph shows that graph of the function f(x)=1−log5(x−2) .
(c)
To determine
The range of any asymptotes from the graph of f , where f(x)=1−log5(x−2) .
(c)
Expert Solution
Answer to Problem 13CT
Solution:
1) The range of the function f(x)=1−log5(x−2) is (−∞,∞) .
2) The vertical asymptote of f(x)=1−log5(x−2) is x=2 .
Explanation of Solution
Given information:
The function is f(x)=1−log5(x−2) .
The domain of the function f(x)=1−log5(x−2) is (2,∞) .
Range of the function is the set of values of the dependent variable for which a function is defined.
From the graph, range of f(x)=1−log5(x−2) is (−∞,∞) .
The graph of f(x)=1−log5(x−2) has no horizontal asymptote.
The vertical asymptote is x=2 .
Therefore, the vertical asymptote of f(x)=1−log5(x−2) is x=2 .
(d)
To determine
The inverse of f , where f(x)=1−log5(x−2) .
(d)
Expert Solution
Answer to Problem 13CT
Solution:
The inverse of f(x)=1−log5(x−2) is f−1(x)=5−x+1+2 .
Explanation of Solution
Given information:
The function is f(x)=1−log5(x−2) .
Consider the function f(x)=1−log5(x−2) .
Step 1: Replace f(x) with y ,
⇒y=1−log5(x−2)
Step 2: Interchange the variables x and y .
Interchange the variables x and y in y=1−log5(x−2) to obtain x=1−log5(y−2) .
Step 3: Solve for y to obtain explicit form of f−1 .
Subtract 1 from both sides,
⇒x−1=−log5(y−2)
Multiplying both sides by −1 ,
⇒−x+1=log5(y−2)
By using, y=logax if and only if ay=xa>0a≠1 ,
⇒5−x+1=y−2
Adding 2 to both sides,
⇒y=5−x+1+2
Replace y by f−1(x) ,
⇒f−1(x)=5−x+1+2
Therefore, the inverse of f(x)=1−log5(x−2) is f−1(x)=5−x+1+2 .
(e)
To determine
The domain and range of f−1 .
(e)
Expert Solution
Answer to Problem 13CT
Solution:
The domain of f−1(x) is (−∞∞), and range of f−1(x) is (2,∞) .
Explanation of Solution
Given information:
The function is f(x)=1−log5(x−2) .
From part (d), f−1(x)=5−x+1+2 .
f−1 is an exponential function.
The exponential function is defined for all real numbers.
The domain of f−1 is (−∞∞) .
The domain of the function f(x)=1−log5(x−2) is (2,∞) .
By using, domain of f(x) = range of f−1(x) .
The range of f−1(x) is (2,∞) .
Therefore, the domain of f−1(x) is (−∞∞) , and range of f−1(x) is (2,∞) .
(e)
To determine
To graph: The function f−1 .
(e)
Expert Solution
Explanation of Solution
Given information:
The function is f(x)=1−log5(x−2) .
Graph:
From part (d), f−1(x)=5−x+1+2 .
Use the steps below to graph the function using graphing calculator:
Step I: Press the ON key.
Step II: Now, press [Y=]. Input the right hand side of the function y=5−x+1+2 in Y1 .
Step III: Then hit [Graph] key to view the graph.
The graph of the function f(x)=5−x+1+2 is as follows:
Interpretation:
By looking at the graph, it is observed that as x→∞ , y→2 . The above graph shows that graph of the function f−1(x)=5−x+1+2 .
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