QuestionFind the domain restriction for the function $f(x) = \frac{1}{x-3}$ . Explain your reasoning.

Studdy Solution

STEP 1

Assumptions1. The composition of two functions is given as $\frac{1}{x-3}$ . We need to find the domain restriction of this function

STEP 2

The domain of a function is the set of all possible input values (x-values) which will output real numbers. In other words, the domain is the set of all real x that make the function "work" and output real y.

STEP 3

For the given function $\frac{1}{x-3}$ , the denominator cannot be zero because division by zero is undefined in mathematics.

STEP 4

To find the domain restriction, we need to find the x-value that makes the denominator zero.
Set the denominator equal to zero and solve for x.
$x-3 =0$

STEP 5

olving the equation gives us the value of x that is not in the domain.
$x =3$

STEP 6

So, the domain of the function $\frac{1}{x-3}$ is all real numbers except3. In interval notation, this can be written as $(-\infty,3) \cup (3, \infty)$ .
The domain restriction for the function is x ≠3.

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QuestionFind the domain restriction for the function $f(x) = \frac{1}{x-3}$ . Explain your reasoning.

Studdy Solution

STEP 1

Assumptions1. The composition of two functions is given as $\frac{1}{x-3}$ . We need to find the domain restriction of this function

STEP 2

The domain of a function is the set of all possible input values (x-values) which will output real numbers. In other words, the domain is the set of all real x that make the function "work" and output real y.

STEP 3

For the given function $\frac{1}{x-3}$ , the denominator cannot be zero because division by zero is undefined in mathematics.

STEP 4

To find the domain restriction, we need to find the x-value that makes the denominator zero.
Set the denominator equal to zero and solve for x.
$x-3 =0$

STEP 5

olving the equation gives us the value of x that is not in the domain.
$x =3$

STEP 6

So, the domain of the function $\frac{1}{x-3}$ is all real numbers except3. In interval notation, this can be written as $(-\infty,3) \cup (3, \infty)$ .
The domain restriction for the function is x ≠3.

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QuestionFind the domain restriction for the function f(x)=1x−3f(x) = \frac{1}{x-3}f(x)=x−31​. Explain your reasoning.

Studdy Solution

STEP 1

Assumptions1. The composition of two functions is given as 1x−3\frac{1}{x-3}x−31​ . We need to find the domain restriction of this function

STEP 2

The domain of a function is the set of all possible input values (x-values) which will output real numbers. In other words, the domain is the set of all real x that make the function "work" and output real y.

STEP 3

For the given function 1x−3\frac{1}{x-3}x−31​, the denominator cannot be zero because division by zero is undefined in mathematics.

STEP 4

To find the domain restriction, we need to find the x-value that makes the denominator zero.Set the denominator equal to zero and solve for x.x−3=0x-3 =0x−3=0

STEP 5

olving the equation gives us the value of x that is not in the domain.x=3x =3x=3

STEP 6

So, the domain of the function 1x−3\frac{1}{x-3}x−31​ is all real numbers except3. In interval notation, this can be written as (−∞,3)∪(3,∞)(-\infty,3) \cup (3, \infty)(−∞,3)∪(3,∞).The domain restriction for the function is x ≠3.

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QuestionFind the domain restriction for the function $f(x) = \frac{1}{x-3}$ . Explain your reasoning.

Assumptions1. The composition of two functions is given as $\frac{1}{x-3}$ . We need to find the domain restriction of this function

For the given function $\frac{1}{x-3}$ , the denominator cannot be zero because division by zero is undefined in mathematics.

To find the domain restriction, we need to find the x-value that makes the denominator zero.
Set the denominator equal to zero and solve for x.
$x-3 =0$

olving the equation gives us the value of x that is not in the domain.
$x =3$

So, the domain of the function $\frac{1}{x-3}$ is all real numbers except3. In interval notation, this can be written as $(-\infty,3) \cup (3, \infty)$ .
The domain restriction for the function is x ≠3.