QuestionWhich statement is true as $x$ approaches $\infty$ for a linear function through the origin?
A. Linear exceeds exponential. B. Exponential exceeds linear. C. Linear exceeds if initial value is greater. D. Exponential exceeds if initial value is greater.

Studdy Solution

STEP 1

Assumptions1. We are comparing a linear function and an exponential function. . The linear function passes through the origin.
3. We are investigating the behavior of these functions as $x$ approaches $\infty$ .

STEP 2

Let's represent the linear function as $f(x) = mx$ where $m$ is the slope of the line and the exponential function as $g(x) = a^x$ where $a$ is a positive constant.

STEP 3

As $x$ approaches $\infty$ , the linear function $f(x)$ will also approach $\infty$ because it is directly proportional to $x$ .

STEP 4

As $x$ approaches $\infty$ , the exponential function $g(x)$ will grow much faster than the linear function $f(x)$ , because the exponential function's growth rate is proportional to its current value, not to $x$ .

STEP 5

Therefore, regardless of the initial values or the slope of the linear function, the exponential function will eventually exceed the linear function as $x$ approaches $\infty$ .
The correct answer is B. The exponential function will eventually exceed the linear function.

Was this helpful?

QuestionWhich statement is true as $x$ approaches $\infty$ for a linear function through the origin?
A. Linear exceeds exponential. B. Exponential exceeds linear. C. Linear exceeds if initial value is greater. D. Exponential exceeds if initial value is greater.

Studdy Solution

STEP 1

Assumptions1. We are comparing a linear function and an exponential function. . The linear function passes through the origin.
3. We are investigating the behavior of these functions as $x$ approaches $\infty$ .

STEP 2

Let's represent the linear function as $f(x) = mx$ where $m$ is the slope of the line and the exponential function as $g(x) = a^x$ where $a$ is a positive constant.

STEP 3

As $x$ approaches $\infty$ , the linear function $f(x)$ will also approach $\infty$ because it is directly proportional to $x$ .

STEP 4

As $x$ approaches $\infty$ , the exponential function $g(x)$ will grow much faster than the linear function $f(x)$ , because the exponential function's growth rate is proportional to its current value, not to $x$ .

STEP 5

Therefore, regardless of the initial values or the slope of the linear function, the exponential function will eventually exceed the linear function as $x$ approaches $\infty$ .
The correct answer is B. The exponential function will eventually exceed the linear function.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionWhich statement is true as xxx approaches ∞\infty∞ for a linear function through the origin?A. Linear exceeds exponential. B. Exponential exceeds linear. C. Linear exceeds if initial value is greater. D. Exponential exceeds if initial value is greater.

Studdy Solution

STEP 1

Assumptions1. We are comparing a linear function and an exponential function. . The linear function passes through the origin.3. We are investigating the behavior of these functions as xxx approaches ∞\infty∞.

STEP 2

Let's represent the linear function as f(x)=mxf(x) = mxf(x)=mx where mmm is the slope of the line and the exponential function as g(x)=axg(x) = a^xg(x)=ax where aaa is a positive constant.

STEP 3

As xxx approaches ∞\infty∞, the linear function f(x)f(x)f(x) will also approach ∞\infty∞ because it is directly proportional to xxx.

STEP 4

As xxx approaches ∞\infty∞, the exponential function g(x)g(x)g(x) will grow much faster than the linear function f(x)f(x)f(x), because the exponential function's growth rate is proportional to its current value, not to xxx.

STEP 5

Therefore, regardless of the initial values or the slope of the linear function, the exponential function will eventually exceed the linear function as xxx approaches ∞\infty∞.The correct answer is B. The exponential function will eventually exceed the linear function.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionWhich statement is true as $x$ approaches $\infty$ for a linear function through the origin?
A. Linear exceeds exponential. B. Exponential exceeds linear. C. Linear exceeds if initial value is greater. D. Exponential exceeds if initial value is greater.

Assumptions1. We are comparing a linear function and an exponential function. . The linear function passes through the origin.
3. We are investigating the behavior of these functions as $x$ approaches $\infty$ .

Let's represent the linear function as $f(x) = mx$ where $m$ is the slope of the line and the exponential function as $g(x) = a^x$ where $a$ is a positive constant.

As $x$ approaches $\infty$ , the linear function $f(x)$ will also approach $\infty$ because it is directly proportional to $x$ .

As $x$ approaches $\infty$ , the exponential function $g(x)$ will grow much faster than the linear function $f(x)$ , because the exponential function's growth rate is proportional to its current value, not to $x$ .

Therefore, regardless of the initial values or the slope of the linear function, the exponential function will eventually exceed the linear function as $x$ approaches $\infty$ .
The correct answer is B. The exponential function will eventually exceed the linear function.