Math  /  Algebra

QuestionWhich statement is true?
The growth rate of the exponential function exceeds the growth rite of the linear function for 1x3-1 \leq x \leq 3.
The growth rate of the exponential function exceeds the growth rate of the linear function for 0x30 \leq x \leq 3. The growth rate of the exponential function exceeds the growth rate of the linear function for 1x1-1 \leq x \leq 1 - The growth rate of the exponential function exceeds the growth rate of the linear function for 1x31 \leq x \leq 3. Mark this and return Save and Exit Next

Studdy Solution

STEP 1

1. We are comparing the growth rates of an exponential function and a linear function.
2. The exponential function is of the form f(x)=abx f(x) = a \cdot b^x where a>0 a > 0 and b>1 b > 1 .
3. The linear function is of the form g(x)=mx+c g(x) = mx + c where m m is the slope.

STEP 2

1. Understand the growth behavior of exponential and linear functions.
2. Compare the growth rates over the specified intervals.
3. Determine the interval where the exponential growth rate exceeds the linear growth rate.

STEP 3

Exponential functions grow at a rate proportional to their current value, meaning they increase rapidly as x x increases.
Linear functions grow at a constant rate, determined by their slope m m .

STEP 4

For the interval 1x3 -1 \leq x \leq 3 , evaluate the growth behavior of both functions. The exponential function grows slowly initially but accelerates as x x increases, while the linear function grows at a constant rate.

STEP 5

For the interval 0x3 0 \leq x \leq 3 , the exponential function's growth rate starts to exceed the linear function's growth rate as x x becomes positive and increases.

STEP 6

For the interval 1x1 -1 \leq x \leq 1 , the exponential function grows slower initially, but by x=1 x = 1 , it starts to exceed the linear function's growth rate.

STEP 7

For the interval 1x3 1 \leq x \leq 3 , the exponential function's growth rate is significantly higher than the linear function's growth rate.

STEP 8

The exponential function's growth rate exceeds the linear function's growth rate for 1x3 1 \leq x \leq 3 .
The correct statement is:
"The growth rate of the exponential function exceeds the growth rate of the linear function for 1x3 1 \leq x \leq 3 ."

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