QuestionFind the domain and range of $F(x)=\frac{1}{{(x-8)}^2}$ . Choose correct options for both.

Studdy Solution

STEP 1

Assumptions1. The function is (x)=\frac{1}{{(x-8)}^}

STEP 2

The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function.

STEP 3

For the function $(x)=\frac{1}{{(x-8)}^2}$ , the denominator cannot be equal to zero because division by zero is undefined in mathematics.

STEP 4

So, we set the denominator equal to zero and solve for x.
$(x-8)^2 =0$

STEP 5

The solution to this equation is $x=8$ .

STEP 6

Therefore, the domain of the function is all real numbers except for $x=8$ . This corresponds to option C The domain of the given function is $\{x \mid x$ is a real number, $x \neq8\}$ .

STEP 7

The range of a function is the set of all possible output values (often the "y" variable), which are the result of the function.

STEP 8

For the function $(x)=\frac{1}{{(x-8)}^2}$ , the output will always be positive because the square of any real number is always positive or zero, and we are dividing1 by this positive number.

STEP 9

However, the function will never equal zero because we are dividing by a positive number, and division by a positive number will never result in zero.

STEP 10

Therefore, the range of the function is all positive real numbers, or $\{y \mid y$ is a real number, $y>0\}$ . This corresponds to option A The range of the given function is $\{y \mid y$ is a real number, $y>0\}$ .
So, the domain of the function $(x)=\frac{}{{(x-8)}^2}$ is all real numbers except for $x=8$ , and the range is all positive real numbers.

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QuestionFind the domain and range of $F(x)=\frac{1}{{(x-8)}^2}$ . Choose correct options for both.

Studdy Solution

STEP 1

Assumptions1. The function is (x)=\frac{1}{{(x-8)}^}

STEP 2

The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function.

STEP 3

For the function $(x)=\frac{1}{{(x-8)}^2}$ , the denominator cannot be equal to zero because division by zero is undefined in mathematics.

STEP 4

So, we set the denominator equal to zero and solve for x.
$(x-8)^2 =0$

STEP 5

The solution to this equation is $x=8$ .

STEP 6

Therefore, the domain of the function is all real numbers except for $x=8$ . This corresponds to option C The domain of the given function is $\{x \mid x$ is a real number, $x \neq8\}$ .

STEP 7

The range of a function is the set of all possible output values (often the "y" variable), which are the result of the function.

STEP 8

For the function $(x)=\frac{1}{{(x-8)}^2}$ , the output will always be positive because the square of any real number is always positive or zero, and we are dividing1 by this positive number.

STEP 9

However, the function will never equal zero because we are dividing by a positive number, and division by a positive number will never result in zero.

STEP 10

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QuestionFind the domain and range of F(x)=1(x−8)2F(x)=\frac{1}{{(x-8)}^2}F(x)=(x−8)21​. Choose correct options for both.

Studdy Solution

STEP 1

Assumptions1. The function is (x)=\frac{1}{{(x-8)}^}

STEP 2

The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function.

STEP 3

For the function (x)=1(x−8)2(x)=\frac{1}{{(x-8)}^2}(x)=(x−8)21​, the denominator cannot be equal to zero because division by zero is undefined in mathematics.

STEP 4

So, we set the denominator equal to zero and solve for x.(x−8)2=0(x-8)^2 =0(x−8)2=0

STEP 5

The solution to this equation is x=8x=8x=8.

STEP 6

Therefore, the domain of the function is all real numbers except for x=8x=8x=8. This corresponds to option C The domain of the given function is {x∣x\{x \mid x{x∣x is a real number, x≠8}x \neq8\}x=8}.

STEP 7

The range of a function is the set of all possible output values (often the "y" variable), which are the result of the function.

STEP 8

For the function (x)=1(x−8)2(x)=\frac{1}{{(x-8)}^2}(x)=(x−8)21​, the output will always be positive because the square of any real number is always positive or zero, and we are dividing1 by this positive number.

STEP 9

However, the function will never equal zero because we are dividing by a positive number, and division by a positive number will never result in zero.

STEP 10

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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.

QuestionFind the domain and range of $F(x)=\frac{1}{{(x-8)}^2}$ . Choose correct options for both.

For the function $(x)=\frac{1}{{(x-8)}^2}$ , the denominator cannot be equal to zero because division by zero is undefined in mathematics.

So, we set the denominator equal to zero and solve for x.
$(x-8)^2 =0$

The solution to this equation is $x=8$ .

Therefore, the domain of the function is all real numbers except for $x=8$ . This corresponds to option C The domain of the given function is $\{x \mid x$ is a real number, $x \neq8\}$ .

For the function $(x)=\frac{1}{{(x-8)}^2}$ , the output will always be positive because the square of any real number is always positive or zero, and we are dividing1 by this positive number.