Answer:
g(x) = f(x + 2)
Step-by-step explanation:
Let's first start by finding the slope of f(x) given the two coordinate points. We are given the y-intercept (0,1) so we can get the equation of f(x). From there, we can use the table values for g, and figure out which transformation will give us the values on the table.
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[tex]\textbf{Solving for f(x):}[/tex]
[tex]\text{We are given coordinates (0,1) and (1,3)}:[/tex]
[tex]\boxed{\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}}[/tex]
Plug in the coordinates:
[tex]\text{Slope = }\frac{3-1}{1-0} = \boxed{2}[/tex]
That means that the slope of f(x) is 2. From here we plug our information into the slope-intercept form of a line to get our equation.
[tex]\textbf{Slope Intercept Form:} ~ y = mx+b[/tex]
Where b represents y-intercept and m represents slope. We are given the y-intercept(when x = 0) in the problem so now let's plug that in with the slope.
[tex]\therefore \boxed{f(x) = 2x+1}[/tex]
[tex]\hrulefill[/tex]
[tex]\textbf{Solving for g(x):}[/tex]
We are given a table, so we can pick out two coordinate points to find the slope.
Let's use (-1,3) and (0,5):
[tex]\boxed{\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}}[/tex]
[tex]\text{Slope = }\frac{5-3}{0-(-1)} = \boxed{2}[/tex]
Now we know that the slope of g(x) is the same as f(x).
The y-intercept of g(x) is 5 since when x is 0, g(x) is 5. We now have all the information for slope-intercept.
[tex]\textbf{Slope Intercept Form:} ~ y = mx+b[/tex]
[tex]\therefore \boxed{g(x) = 2x+5}[/tex]
Now we need to see which answer option will make g(x) = f(x).
For g(x) = f(x-2), plug in (x-2) where there is x in f(x):
[tex]f(x-2) = 2(x-2)+1\\f(x-2) = 2x-4+1\\f(x-2) = 2x-3[/tex]
Option 1 is incorrect since f(x+2) ≠ g(x)
For g(x) = f(x+2), plug in (x+2) where there is x in f(x):
[tex]f(x+2) = 2(x+2)+1\\f(x+2) = 2x+4+1\\f(x+2) = 2x+5[/tex]
Since f(x+2) = g(x) that is the correct answer!
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