Evaluate the logarithmic function for three values of x between zero and one, three values of x greater than 1, and at x=1. Show your work. Use the resulting ordered pairs to plot the graph. State the domain and the range of the function.
g(x) = ln x + 2
To evaluate the function g(x) = ln x + 2 for three values of x between zero and one, three values of x greater than 1, and at x=1, we can substitute these values into the function and calculate the corresponding values of g(x).
For values of x between zero and one:
1) Let x = 0.25:
g(0.25) = ln(0.25) + 2
Using a calculator, ln(0.25) ≈ -1.3863
g(0.25) ≈ -1.3863 + 2 ≈ 0.6137
So, the first ordered pair is (0.25, 0.6137).
2) Let x = 0.5:
g(0.5) = ln(0.5) + 2
Using a calculator, ln(0.5) ≈ -0.6931
g(0.5) ≈ -0.6931 + 2 ≈ 1.3069
The second ordered pair is (0.5, 1.3069).
3) Let x = 0.75:
g(0.75) = ln(0.75) + 2
Using a calculator, ln(0.75) ≈ -0.2877
g(0.75) ≈ -0.2877 + 2 ≈ 1.7123
The third ordered pair is (0.75, 1.7123).
For values of x greater than 1:
1) Let x = 2:
g(2) = ln(2) + 2
Using a calculator, ln(2) ≈ 0.6931
g(2) ≈ 0.6931 + 2 ≈ 2.6931
The fourth ordered pair is (2, 2.6931).
2) Let x = 5:
g(5) = ln(5) + 2
Using a calculator, ln(5) ≈ 1.6094
g(5) ≈ 1.6094 + 2 ≈ 3.6094
The fifth ordered pair is (5, 3.6094).
3) Let x = 10:
g(10) = ln(10) + 2
Using a calculator, ln(10) ≈ 2.3026
g(10) ≈ 2.3026 + 2 ≈ 4.3026
The sixth ordered pair is (10, 4.3026).
Finally, at x = 1:
g(1) = ln(1) + 2
Since ln(1) = 0, g(1) = 0 + 2 = 2
The seventh ordered pair is (1, 2).
Now, we have the following ordered pairs:
(0.25, 0.6137), (0.5, 1.3069), (0.75, 1.7123), (2, 2.6931), (5, 3.6094), (10, 4.3026), (1, 2).
Plotting these points on a graph, we get:
|
5 +
|
|
|
|
|
4 +
|
|
|
3 + *
|
|
|
2 + *
|
|
1 +
|
+------------------------------------
0.25 0.5 0.75 1 2 5 10
The domain of the function g(x) = ln x + 2 is the set of all real numbers such that x > 0. In this case, the domain is (0, ∞).
The range of the function g(x) = ln x + 2 is the set of all real numbers. In this case, the range is (-∞, ∞).
To evaluate the logarithmic function g(x) = ln(x) + 2 for three values of x between zero and one, three values of x greater than 1, and at x = 1, we can follow these steps:
Step 1: Calculate the function values for three values of x between zero and one.
Let's choose x = 0.1, x = 0.5, and x = 0.9.
For x = 0.1:
g(0.1) = ln(0.1) + 2
To evaluate the natural logarithm for x < 1, we can use the rule:
ln(x) = -ln(1/x)
g(0.1) = -ln(1/0.1) + 2
= -ln(10) + 2
Using a calculator or mathematical software, we find:
g(0.1) ≈ -2.3026 + 2
≈ -0.3026
So, one ordered pair is (0.1, -0.3026).
Similarly, for x = 0.5:
g(0.5) = ln(0.5) + 2
≈ -0.6931 + 2
≈ 1.3069
So, another ordered pair is (0.5, 1.3069).
And for x = 0.9:
g(0.9) = ln(0.9) + 2
≈ -0.1054 + 2
≈ 1.8946
So, the third ordered pair is (0.9, 1.8946).
Step 2: Calculate the function values for three values of x greater than 1.
Let's choose x = 2, x = 5, and x = 10.
For x = 2:
g(2) = ln(2) + 2
Using a calculator or mathematical software:
g(2) ≈ 0.6931 + 2
≈ 2.6931
So, the fourth ordered pair is (2, 2.6931).
Similarly, for x = 5:
g(5) = ln(5) + 2
≈ 1.6094 + 2
≈ 3.6094
So, another ordered pair is (5, 3.6094).
And for x = 10:
g(10) = ln(10) + 2
≈ 2.3026 + 2
≈ 4.3026
So, the sixth ordered pair is (10, 4.3026).
Step 3: Calculate the function value for x = 1.
For x = 1:
g(1) = ln(1) + 2
The natural logarithm of 1 is 0, so:
g(1) = 0 + 2
= 2
So, the seventh ordered pair is (1, 2).
Plotting the points on a graph, we get:
^
|
4 |- * (10, 4.3026)
|
3 |- * (5, 3.6094)
|
2 |- * (2, 2.6931)
|
1 |- * (1, 2)
|
0 |- * (0.9, 1.8946) * (0.5, 1.3069) * (0.1, -0.3026)
----------------------------------------------->
0 1 2 3 4 5 6 7 8 9 10
The domain of the function is all positive real numbers, excluding zero. So, (0, ∞).
The range of the function is all real numbers, since the natural logarithm is defined for positive numbers. So, (-∞, ∞).