Answer :
Let's go step-by-step through each part of the question.
a. Plot the data for the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] on a grid:
We have the following data points for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & \frac{1}{16} & \frac{1}{4} & 1 & 4 & 16 \\ \hline \end{array} \][/tex]
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 & 3 \\ \hline g(x) & 3 & 4 & 5 & 6 & 7 \\ \hline \end{array} \][/tex]
Plotting these points on a grid:
![Function Plot]()
b. Identify each function as linear, quadratic, or exponential, and use complete sentences to explain your choices.
- Function [tex]\(f(x)\)[/tex]:
The function values [tex]\(\frac{1}{16}, \frac{1}{4}, 1, 4, 16\)[/tex] can be written as [tex]\(f(x) = 4^x\)[/tex]. This function represents an exponential relationship because the function values increase by a factor of 4 each time [tex]\(x\)[/tex] increases by 1 unit. In other words, the rate of increase in the function values happens exponentially.
- Function [tex]\(g(x)\)[/tex]:
The function values [tex]\(3, 4, 5, 6, 7\)[/tex] follow the pattern of [tex]\(g(x) = x + 4\)[/tex]. This function represents a linear relationship because the function values increase by the same amount (1 unit) for each increment in [tex]\(x\)[/tex] by 1 unit.
c. Describe what happens to the function values in each function as [tex]\(x\)[/tex] increases from left to right.
- For [tex]\(f(x)\)[/tex]: As [tex]\(x\)[/tex] increases, the function values increase exponentially. This means even small changes in [tex]\(x\)[/tex] lead to large changes in the function values. The graph of [tex]\(f(x)\)[/tex] shows a rapid increase for positive [tex]\(x\)[/tex] and a rapid decrease for negative [tex]\(x\)[/tex].
- For [tex]\(g(x)\)[/tex]: As [tex]\(x\)[/tex] increases, the function values increase linearly. This means that the function values change by the same amount for each unit increase in [tex]\(x\)[/tex]. The graph of [tex]\(g(x)\)[/tex] is a straight line with a consistent upward slope.
d. At what value(s) of [tex]\(x\)[/tex] are the function values equal? If you cannot give exact values for [tex]\(x\)[/tex], give estimates.
To find where [tex]\(f(x) = g(x)\)[/tex], we set [tex]\(4^x = x + 4\)[/tex].
This equation is not straightforward to solve algebraically, but we can estimate by inspecting the plotted graphs or by trial and error:
- For [tex]\(x = 0\)[/tex]: [tex]\(4^0 = 1\)[/tex] and [tex]\(0 + 4 = 4\)[/tex]. They are not equal.
- For [tex]\(x = 1\)[/tex]: [tex]\(4^1 = 4\)[/tex] and [tex]\(1 + 4 = 5\)[/tex]. They are not equal.
- For [tex]\(x = 2\)[/tex]: [tex]\(4^2 = 16\)[/tex] and [tex]\(2 + 4 = 6\)[/tex]. They are not equal.
- For [tex]\(x = -1\)[/tex]: [tex]\(4^{-1} = \frac{1}{4}\)[/tex] and [tex]\(-1 + 4 = 3\)[/tex]. They are not equal.
By inspecting the graphs or exploring further with methods such as numerical solving or graph intersection points, we can see that around [tex]\(x \approx 0.5\)[/tex], the function values might be equal. Therefore, our estimation gives us an intersection point at approximately [tex]\(x \approx 0.5\)[/tex]. This is a rough estimate and the precise intersection would need a more detailed calculation or graphical inspection.
a. Plot the data for the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] on a grid:
We have the following data points for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & \frac{1}{16} & \frac{1}{4} & 1 & 4 & 16 \\ \hline \end{array} \][/tex]
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 & 3 \\ \hline g(x) & 3 & 4 & 5 & 6 & 7 \\ \hline \end{array} \][/tex]
Plotting these points on a grid:
![Function Plot]()
b. Identify each function as linear, quadratic, or exponential, and use complete sentences to explain your choices.
- Function [tex]\(f(x)\)[/tex]:
The function values [tex]\(\frac{1}{16}, \frac{1}{4}, 1, 4, 16\)[/tex] can be written as [tex]\(f(x) = 4^x\)[/tex]. This function represents an exponential relationship because the function values increase by a factor of 4 each time [tex]\(x\)[/tex] increases by 1 unit. In other words, the rate of increase in the function values happens exponentially.
- Function [tex]\(g(x)\)[/tex]:
The function values [tex]\(3, 4, 5, 6, 7\)[/tex] follow the pattern of [tex]\(g(x) = x + 4\)[/tex]. This function represents a linear relationship because the function values increase by the same amount (1 unit) for each increment in [tex]\(x\)[/tex] by 1 unit.
c. Describe what happens to the function values in each function as [tex]\(x\)[/tex] increases from left to right.
- For [tex]\(f(x)\)[/tex]: As [tex]\(x\)[/tex] increases, the function values increase exponentially. This means even small changes in [tex]\(x\)[/tex] lead to large changes in the function values. The graph of [tex]\(f(x)\)[/tex] shows a rapid increase for positive [tex]\(x\)[/tex] and a rapid decrease for negative [tex]\(x\)[/tex].
- For [tex]\(g(x)\)[/tex]: As [tex]\(x\)[/tex] increases, the function values increase linearly. This means that the function values change by the same amount for each unit increase in [tex]\(x\)[/tex]. The graph of [tex]\(g(x)\)[/tex] is a straight line with a consistent upward slope.
d. At what value(s) of [tex]\(x\)[/tex] are the function values equal? If you cannot give exact values for [tex]\(x\)[/tex], give estimates.
To find where [tex]\(f(x) = g(x)\)[/tex], we set [tex]\(4^x = x + 4\)[/tex].
This equation is not straightforward to solve algebraically, but we can estimate by inspecting the plotted graphs or by trial and error:
- For [tex]\(x = 0\)[/tex]: [tex]\(4^0 = 1\)[/tex] and [tex]\(0 + 4 = 4\)[/tex]. They are not equal.
- For [tex]\(x = 1\)[/tex]: [tex]\(4^1 = 4\)[/tex] and [tex]\(1 + 4 = 5\)[/tex]. They are not equal.
- For [tex]\(x = 2\)[/tex]: [tex]\(4^2 = 16\)[/tex] and [tex]\(2 + 4 = 6\)[/tex]. They are not equal.
- For [tex]\(x = -1\)[/tex]: [tex]\(4^{-1} = \frac{1}{4}\)[/tex] and [tex]\(-1 + 4 = 3\)[/tex]. They are not equal.
By inspecting the graphs or exploring further with methods such as numerical solving or graph intersection points, we can see that around [tex]\(x \approx 0.5\)[/tex], the function values might be equal. Therefore, our estimation gives us an intersection point at approximately [tex]\(x \approx 0.5\)[/tex]. This is a rough estimate and the precise intersection would need a more detailed calculation or graphical inspection.
