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To determine which exponential function grows at a faster rate than the quadratic function for the range [tex]\(0 < x < 3\)[/tex], let's analyze the behavior of both functions across this interval.
First, we need to identify key points for both the quadratic and the potential exponential functions.
Here's the given table for the quadratic function:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ \hline 1 & 3 \\ \hline 2 & 12 \\ \hline 3 & 27 \\ \hline \end{array} \][/tex]
These points suggest that the quadratic function is of the form [tex]\( y = x^3 \)[/tex].
Next, let's consider an exponential function of the form [tex]\( y = a \cdot b^x \)[/tex]. A simple and common choice for exponential functions is [tex]\( y = 2^x \)[/tex].
We will now compare the values of the exponential function [tex]\( y = 2^x \)[/tex] with the values of the quadratic function [tex]\( y = x^3 \)[/tex] for [tex]\(x = 0, 1, 2, 3\)[/tex]:
1. For [tex]\(x = 0\)[/tex]:
- Quadratic value: [tex]\(0^3 = 0\)[/tex]
- Exponential value: [tex]\(2^0 = 1\)[/tex]
2. For [tex]\(x = 1\)[/tex]:
- Quadratic value: [tex]\(1^3 = 1\)[/tex]
- Exponential value: [tex]\(2^1 = 2\)[/tex]
3. For [tex]\(x = 2\)[/tex]:
- Quadratic value: [tex]\(2^3 = 8\)[/tex]
- Exponential value: [tex]\(2^2 = 4\)[/tex]
4. For [tex]\(x = 3\)[/tex]:
- Quadratic value: [tex]\(3^3 = 27\)[/tex]
- Exponential value: [tex]\(2^3 = 8\)[/tex]
To summarize, the values are:
[tex]\[ \begin{array}{|c|c|c|} \hline x & \text{Quadratic Function } (x^3) & \text{Exponential Function } (2^x) \\ \hline 0 & 0 & 1 \\ \hline 1 & 1 & 2 \\ \hline 2 & 8 & 4 \\ \hline 3 & 27 & 8 \\ \hline \end{array} \][/tex]
From this comparison, we observe:
- At [tex]\(x = 0\)[/tex], the exponential function starts at 1, whereas the quadratic starts at 0.
- At [tex]\(x = 1\)[/tex], the exponential function value (2) is greater than the quadratic value (1).
- At [tex]\(x = 2\)[/tex], the quadratic function value (8) exceeds the exponential function value (4).
- At [tex]\(x = 3\)[/tex], the quadratic function value (27) exceeds the exponential function value (8).
It's clear that the exponential function [tex]\(2^x\)[/tex] grows faster than the quadratic function [tex]\(x^3\)[/tex] for the interval [tex]\(0 < x < 3\)[/tex] but does not grow faster at [tex]\(x = 2\)[/tex] and [tex]\(x = 3\)[/tex].
Thus, an example of an exponential function that grows faster than the quadratic function [tex]\(y = x^3\)[/tex] within the interval [tex]\(0 < x < 3\)[/tex] is [tex]\(y = 2^x\)[/tex].
First, we need to identify key points for both the quadratic and the potential exponential functions.
Here's the given table for the quadratic function:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ \hline 1 & 3 \\ \hline 2 & 12 \\ \hline 3 & 27 \\ \hline \end{array} \][/tex]
These points suggest that the quadratic function is of the form [tex]\( y = x^3 \)[/tex].
Next, let's consider an exponential function of the form [tex]\( y = a \cdot b^x \)[/tex]. A simple and common choice for exponential functions is [tex]\( y = 2^x \)[/tex].
We will now compare the values of the exponential function [tex]\( y = 2^x \)[/tex] with the values of the quadratic function [tex]\( y = x^3 \)[/tex] for [tex]\(x = 0, 1, 2, 3\)[/tex]:
1. For [tex]\(x = 0\)[/tex]:
- Quadratic value: [tex]\(0^3 = 0\)[/tex]
- Exponential value: [tex]\(2^0 = 1\)[/tex]
2. For [tex]\(x = 1\)[/tex]:
- Quadratic value: [tex]\(1^3 = 1\)[/tex]
- Exponential value: [tex]\(2^1 = 2\)[/tex]
3. For [tex]\(x = 2\)[/tex]:
- Quadratic value: [tex]\(2^3 = 8\)[/tex]
- Exponential value: [tex]\(2^2 = 4\)[/tex]
4. For [tex]\(x = 3\)[/tex]:
- Quadratic value: [tex]\(3^3 = 27\)[/tex]
- Exponential value: [tex]\(2^3 = 8\)[/tex]
To summarize, the values are:
[tex]\[ \begin{array}{|c|c|c|} \hline x & \text{Quadratic Function } (x^3) & \text{Exponential Function } (2^x) \\ \hline 0 & 0 & 1 \\ \hline 1 & 1 & 2 \\ \hline 2 & 8 & 4 \\ \hline 3 & 27 & 8 \\ \hline \end{array} \][/tex]
From this comparison, we observe:
- At [tex]\(x = 0\)[/tex], the exponential function starts at 1, whereas the quadratic starts at 0.
- At [tex]\(x = 1\)[/tex], the exponential function value (2) is greater than the quadratic value (1).
- At [tex]\(x = 2\)[/tex], the quadratic function value (8) exceeds the exponential function value (4).
- At [tex]\(x = 3\)[/tex], the quadratic function value (27) exceeds the exponential function value (8).
It's clear that the exponential function [tex]\(2^x\)[/tex] grows faster than the quadratic function [tex]\(x^3\)[/tex] for the interval [tex]\(0 < x < 3\)[/tex] but does not grow faster at [tex]\(x = 2\)[/tex] and [tex]\(x = 3\)[/tex].
Thus, an example of an exponential function that grows faster than the quadratic function [tex]\(y = x^3\)[/tex] within the interval [tex]\(0 < x < 3\)[/tex] is [tex]\(y = 2^x\)[/tex].
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