Question $f$ is an odd, continuous, periodic function with period 2. If $g(x) = \int_{0}^{1} f(t) dt$ , then find properties of $g(x)$ .

Studdy Solution

STEP 1

Assumptions1. The function $f$ is odd, which means $f(-x) = -f(x)$ for all $x$ . . The function $f$ is continuous for all values of $x$ .
3. The function $f$ is periodic with period, which means $f(x+) = f(x)$ for all $x$ .
4. The function $g(x)$ is defined as the integral of $f(t)$ from0 to1, $g(x) = \int_{0}^{1} f(t) dt$ .

STEP 2

First, we need to understand the properties of an odd function. An odd function is symmetric about the origin. This means that $f(-x) = -f(x)$ for all $x$ .

STEP 3

Next, we need to understand the properties of a periodic function. A function is said to be periodic with period $$ if $f(x+) = f(x)$ for all $x$. In this case, $f$ is periodic with period2.

STEP 4

The function $g(x)$ is defined as the integral of $f(t)$ from0 to1. This means that $g(x)$ is a constant function, as it does not depend on $x$ .

STEP 5

Since $g(x)$ is a constant function, it is neither odd nor even. Therefore, option (1) is incorrect.

STEP 6

Since $g(x)$ is a constant function, its value does not change with $x$ . Therefore, $g(n)$ is the same as $g(x)$ for any $n \in N$ . Since we do not know the value of the integral $\int_{0}^{1} f(t) dt$ , we cannot say that $g(n)=0$ for $n \in N$ . Therefore, option (2) is incorrect.

STEP 7

Similarly, $g(2n)$ is the same as $g(x)$ for any $n \in N$ . Therefore, we cannot say that $g(2n)=0$ for $n \in N$ . Therefore, option (3) is incorrect.

STEP 8

Since $g(x)$ is a constant function, it does not have a period. Therefore, $g(x)$ is non-periodic. Therefore, option (4) is correct.
The correct answer is option (4): $g(x)$ is non-periodic.

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Question $f$ is an odd, continuous, periodic function with period 2. If $g(x) = \int_{0}^{1} f(t) dt$ , then find properties of $g(x)$ .

Studdy Solution

STEP 1

STEP 2

First, we need to understand the properties of an odd function. An odd function is symmetric about the origin. This means that $f(-x) = -f(x)$ for all $x$ .

STEP 3

Next, we need to understand the properties of a periodic function. A function is said to be periodic with period $$ if $f(x+) = f(x)$ for all $x$. In this case, $f$ is periodic with period2.

STEP 4

The function $g(x)$ is defined as the integral of $f(t)$ from0 to1. This means that $g(x)$ is a constant function, as it does not depend on $x$ .

STEP 5

Since $g(x)$ is a constant function, it is neither odd nor even. Therefore, option (1) is incorrect.

STEP 6

Since $g(x)$ is a constant function, its value does not change with $x$ . Therefore, $g(n)$ is the same as $g(x)$ for any $n \in N$ . Since we do not know the value of the integral $\int_{0}^{1} f(t) dt$ , we cannot say that $g(n)=0$ for $n \in N$ . Therefore, option (2) is incorrect.

STEP 7

Similarly, $g(2n)$ is the same as $g(x)$ for any $n \in N$ . Therefore, we cannot say that $g(2n)=0$ for $n \in N$ . Therefore, option (3) is incorrect.

STEP 8

Since $g(x)$ is a constant function, it does not have a period. Therefore, $g(x)$ is non-periodic. Therefore, option (4) is correct.
The correct answer is option (4): $g(x)$ is non-periodic.

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Questionfff is an odd, continuous, periodic function with period 2. If g(x)=∫01f(t)dtg(x) = \int_{0}^{1} f(t) dtg(x)=∫01​f(t)dt, then find properties of g(x)g(x)g(x).

Studdy Solution

STEP 1

STEP 2

First, we need to understand the properties of an odd function. An odd function is symmetric about the origin. This means that f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x) for all xxx.

STEP 3

Next, we need to understand the properties of a periodic function. A function is said to be periodic with period $$ if $f(x+) = f(x)$ for all $x$. In this case, $f$ is periodic with period2.

STEP 4

The function g(x)g(x)g(x) is defined as the integral of f(t)f(t)f(t) from0 to1. This means that g(x)g(x)g(x) is a constant function, as it does not depend on xxx.

STEP 5

Since g(x)g(x)g(x) is a constant function, it is neither odd nor even. Therefore, option (1) is incorrect.

STEP 6

STEP 7

Similarly, g(2n)g(2n)g(2n) is the same as g(x)g(x)g(x) for any n∈Nn \in Nn∈N. Therefore, we cannot say that g(2n)=0g(2n)=0g(2n)=0 for n∈Nn \in Nn∈N. Therefore, option (3) is incorrect.

STEP 8

Since g(x)g(x)g(x) is a constant function, it does not have a period. Therefore, g(x)g(x)g(x) is non-periodic. Therefore, option (4) is correct.The correct answer is option (4): g(x)g(x)g(x) is non-periodic.

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Question $f$ is an odd, continuous, periodic function with period 2. If $g(x) = \int_{0}^{1} f(t) dt$ , then find properties of $g(x)$ .

First, we need to understand the properties of an odd function. An odd function is symmetric about the origin. This means that $f(-x) = -f(x)$ for all $x$ .

The function $g(x)$ is defined as the integral of $f(t)$ from0 to1. This means that $g(x)$ is a constant function, as it does not depend on $x$ .

Since $g(x)$ is a constant function, it is neither odd nor even. Therefore, option (1) is incorrect.

Similarly, $g(2n)$ is the same as $g(x)$ for any $n \in N$ . Therefore, we cannot say that $g(2n)=0$ for $n \in N$ . Therefore, option (3) is incorrect.

Since $g(x)$ is a constant function, it does not have a period. Therefore, $g(x)$ is non-periodic. Therefore, option (4) is correct.
The correct answer is option (4): $g(x)$ is non-periodic.