Question

the graphs of linear functions f and g are shown below.
what is the solution to f(x) = g(x)?
use the on - screen keyboard to type the correct response in the box below.
5.
a table showing pairs of x - and y - values is shown below.

x	y
1	1.5
2	4.5
3	13.5
4	40.5

which equation describes the relationship between the pairs of x - and y - values?
a. $y=(0.5cdot3)^x$
b. $y = 0.5cdot3^x$
c. $y = 2x-0.5$
d. $y=x + 0.5$

Explanation:

First Sub - Question (Solution to \(f(x)=g(x)\))

Step1: Recall the meaning of \(f(x) = g(x)\)

The solution to \(f(x)=g(x)\) for the graphs of linear functions \(f\) and \(g\) is the \(x\) - coordinate of the point of intersection of the two lines.

Step2: Identify the intersection point

From the graph (even with the partial view), the two lines \(f\) and \(g\) intersect at a point. By looking at the grid, we can see that the \(x\) - coordinate of the intersection point is \(- 2\) (assuming the grid lines and the intersection is at \(x=-2\) from the visual cues of the graph).

Step1: Analyze option A

For option A, \(y=(0.5\times3)^{x}=1.5^{x}\). When \(x = 0\), \(y = 1.5^{0}=1
eq0.5\). So option A is incorrect.

Step2: Analyze option B

For option B, \(y = 0.5\times3^{x}\).

• When \(x = 0\), \(y=0.5\times3^{0}=0.5\times1 = 0.5\) (matches the table).
• When \(x = 1\), \(y=0.5\times3^{1}=0.5\times3 = 1.5\) (matches the table).
• When \(x = 2\), \(y=0.5\times3^{2}=0.5\times9 = 4.5\) (matches the table).
• When \(x = 3\), \(y=0.5\times3^{3}=0.5\times27 = 13.5\) (matches the table).
• When \(x = 4\), \(y=0.5\times3^{4}=0.5\times81 = 40.5\) (matches the table).

Step3: Analyze option C

For option C, \(y = 2x-0.5\). When \(x = 2\), \(y=2\times2 - 0.5=4 - 0.5 = 3.5
eq4.5\). So option C is incorrect.

Step4: Analyze option D

For option D, \(y=x + 0.5\). When \(x = 2\), \(y=2 + 0.5=2.5
eq4.5\). So option D is incorrect.

Answer:

\(-2\)

Second Sub - Question (Equation for the table)