aComparison: The functions have the same domain, but different ranges.
Feature
f(x)
g(x)
Domain
-1 ≤ x ≤ 3
-1 ≤ x ≤ 3
Range
-7 ≤ f(x) ≤ 5
-4 ≤ g(x) ≤ 4
B
b
Feature
f(x)
g(x)
Slope
3
2
y-intercept
-4
-2
Practice makes perfect
a Let's find the domain and range of f(x) first. Then we can find the domain and range of g(x) and compare them.
f(x)
We are told to assume that the domain of f(x) includes all real numbers between the least and the greatest values shown in the table. Since the least value of x is -1 and the greatest is 3, the domain of f(x) is:
-1 ≤ x ≤ 3.The range is the corresponding output values. From the table, we can see that the range is:
-7 ≤ f(x) ≤ 5.
g(x)
We can determine the domain and range of g(x) by looking at the graph. The function starts at x=-1 and ends at x=3. Thus, the domain is all real numbers of x such that
-1 ≤ x ≤ 3.
Once again, the range is the corresponding output values. We have to look at the y-axis to identify the range. We can tell that g(x) starts at g(x)=-4 and ends at g(x)=4. Thus, the range of g(x) is:
-4 ≤ g(x) ≤ 4.
Compare
Let's compare them!
Feature
f(x)
g(x)
Domain
-1 ≤ x ≤ 3
-1 ≤ x ≤ 3
Range
-7 ≤ f(x) ≤ 5
-4 ≤ g(x) ≤ 4
The functions have the same domains! However, the range of f(x) is different than the range of g(x).
b To answer the posed questions, let's first find the rules for f(x) and g(x).
Rule for f(x)
We know that f(x) is a linear function, so we can write it in the slope-intercept form. Let's take two points from the table to find the slope using the Slope Formula! Let's use (-1,-7) and (0, -4).
The slope is 3. From the table, we know that f(0)=-4, so the y-intercept b is -4. Therefore, the final rule for f(x) is:
f(x)=3x-4.
Rule for g(x)
Looking at the graph of g(x), we can tell that it passes through the points (-1,-4) and (3,4). Let's substitute them in the Slope Formula to find the slope!
The slope is 2. From the graph, we can also see that the function intercepts the y-axis at y=-2. Therefore, b=-2, and we can write the final equation for g(x) as
g(x)=2x-2.
Conclusion
Using the rules for the functions, we summarize their key features in the table below.
