Question1 AA. 12 Compare linear, exponential, and quadratic growth 39 V
Both of these functions grow as $x$ gets larger and larger. Which function eventually exceeds the other? $f(x)=\frac{3}{2} x+6 \quad g(x)=\frac{1}{2} x^{2}-x+6$ Submit

Studdy Solution

STEP 1

1. We are comparing the growth of two functions: a linear function $f(x) = \frac{3}{2}x + 6$ and a quadratic function $g(x) = \frac{1}{2}x^2 - x + 6$ .
2. We need to determine which function eventually exceeds the other as $x$ becomes very large.

STEP 2

1. Analyze the growth behavior of the linear function $f(x)$ .
2. Analyze the growth behavior of the quadratic function $g(x)$ .
3. Compare the growth rates of the two functions for large values of $x$ .

STEP 3

The function $f(x) = \frac{3}{2}x + 6$ is linear, which means it grows at a constant rate as $x$ increases.

STEP 4

The function $g(x) = \frac{1}{2}x^2 - x + 6$ is quadratic, which means its growth rate increases as $x$ increases. The leading term $\frac{1}{2}x^2$ dominates the growth for large $x$ .

STEP 5

For very large values of $x$ , the quadratic term $\frac{1}{2}x^2$ in $g(x)$ will grow faster than the linear term $\frac{3}{2}x$ in $f(x)$ . Therefore, $g(x)$ will eventually exceed $f(x)$ .
The function $g(x)$ eventually exceeds $f(x)$ as $x$ becomes very large.

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