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Concept explainers
To graph the function g(x)=|x|+1 and determine its domain and range and its relation with the graph of the function f(x)=|x| .
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Answer to Problem 1CYU
Domain: x∈ℝ
Range: y∈[1,∞)
Explanation of Solution
Given:
Function: g(x)=|x|+1
Concept used:
The general form of an absolute value function is f(x)=a|x−h|+k .
The graph of f(x)=a|x−h|+k is a “V” with vertex (h,k) , slope m=a on the right side of the vertex (x>h) and slope m=−a on the left side of the vertex (x<h) .
The graph of f(x)=−a|x−h|+k is an upside-down “V” with vertex (h,k) , slope m=−a for (x>h) and slope m=a for (x<h) .
Calculation for graph:
Consider g(x)=|x|+1,
Compare with the general form:
g(x)=a|x−h|+k
Here, the value of a=1 , so the graph opens upwards with a slope of 1 .
The value of h=0 and the value of k=1 , so the vertex of the graph is (0,1) .
Values of x | Values of g (x) |
0 | 1 |
1 | 2 |
-1 | 2 |
2 | 3 |
-2 | 3 |
By taking different values of x , the graph can be plotted.
Graph:
The graph of g(x)=|x|+1 :
Interpretation:
From the graph, it can be observed that the horizontal extent of the graph is from −∞ to ∞ , so the domain of g(x) is (−∞,∞) .
The vertical extent of the graph is from 1 to ∞ , so the range of g(x) is [1,∞) .
Comparison:
Now, comparing the graph of g(x) with the graph of f(x)=|x| .
Graph of f(x)=|x|:
The domain of both the graphs is (−∞,∞) .
Therefore, both the graphs have same domain.
Conclusion:
Therefore, the domain of g(x)=|x|+1 is x∈ℝ and range is y∈[1,∞) .
Chapter 3 Solutions
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
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