QuestionFind the domain and range of the function $f(x)=\sqrt{x+5}-3$ in interval and inequality notation.

Studdy Solution

STEP 1

Assumptions1. The function is defined as $f(x)=\sqrt{x+5}-3$ . We need to find the domain and range of the function3. The domain of a function is the set of all possible x-values that will output a real number4. The range of a function is the set of all possible y-values that can be obtained by substituting the x-values in the domain into the function5. The square root of a real number is defined only for numbers greater than or equal to zero

STEP 2

To find the domain of the function, we need to set the expression inside the square root greater than or equal to zero, because the square root of a negative number is not a real number.
$x+5 \geq0$

STEP 3

Now, solve the inequality to find the domain.
$x \geq -5$

STEP 4

So, the domain of the function in inequality notation is $x \geq -$ .

STEP 5

To express the domain in interval notation, we write it as $[-5, \infty)$ . This means that x can be any number greater than or equal to -5.

STEP 6

To find the range of the function, we need to consider the minimum and maximum values that the function can take. The square root function $\sqrt{x}$ always gives non-negative values, and by subtracting3, the minimum value the function can take is -3.

STEP 7

The maximum value of the function is $\infty$ , because as x approaches $\infty$ , the value of $\sqrt{x+5}-3$ also approaches $\infty$ .

STEP 8

So, the range of the function in inequality notation is $y \geq -3$ .

STEP 9

To express the range in interval notation, we write it as $[-3, \infty)$ . This means that y can be any number greater than or equal to -3.
The domain of the function $f(x)=\sqrt{x+5}-3$ is $[-5, \infty)$ in interval notation and $x \geq -5$ in inequality notation. The range of the function is $[-3, \infty)$ in interval notation and $y \geq -3$ in inequality notation.

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QuestionFind the domain and range of the function $f(x)=\sqrt{x+5}-3$ in interval and inequality notation.

Studdy Solution

STEP 1

STEP 2

To find the domain of the function, we need to set the expression inside the square root greater than or equal to zero, because the square root of a negative number is not a real number.
$x+5 \geq0$

STEP 3

Now, solve the inequality to find the domain.
$x \geq -5$

STEP 4

So, the domain of the function in inequality notation is $x \geq -$ .

STEP 5

To express the domain in interval notation, we write it as $[-5, \infty)$ . This means that x can be any number greater than or equal to -5.

STEP 6

To find the range of the function, we need to consider the minimum and maximum values that the function can take. The square root function $\sqrt{x}$ always gives non-negative values, and by subtracting3, the minimum value the function can take is -3.

STEP 7

The maximum value of the function is $\infty$ , because as x approaches $\infty$ , the value of $\sqrt{x+5}-3$ also approaches $\infty$ .

STEP 8

So, the range of the function in inequality notation is $y \geq -3$ .

STEP 9

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QuestionFind the domain and range of the function f(x)=x+5−3f(x)=\sqrt{x+5}-3f(x)=x+5​−3 in interval and inequality notation.

Studdy Solution

STEP 1

STEP 2

To find the domain of the function, we need to set the expression inside the square root greater than or equal to zero, because the square root of a negative number is not a real number.x+5≥0x+5 \geq0x+5≥0

STEP 3

Now, solve the inequality to find the domain.x≥−5x \geq -5x≥−5

STEP 4

So, the domain of the function in inequality notation is x≥−x \geq -x≥−.

STEP 5

To express the domain in interval notation, we write it as [−5,∞)[-5, \infty)[−5,∞). This means that x can be any number greater than or equal to -5.

STEP 6

To find the range of the function, we need to consider the minimum and maximum values that the function can take. The square root function x\sqrt{x}x​ always gives non-negative values, and by subtracting3, the minimum value the function can take is -3.

STEP 7

The maximum value of the function is ∞\infty∞, because as x approaches ∞\infty∞, the value of x+5−3\sqrt{x+5}-3x+5​−3 also approaches ∞\infty∞.

STEP 8

So, the range of the function in inequality notation is y≥−3y \geq -3y≥−3.

STEP 9

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QuestionFind the domain and range of the function $f(x)=\sqrt{x+5}-3$ in interval and inequality notation.

To find the domain of the function, we need to set the expression inside the square root greater than or equal to zero, because the square root of a negative number is not a real number.
$x+5 \geq0$

Now, solve the inequality to find the domain.
$x \geq -5$

So, the domain of the function in inequality notation is $x \geq -$ .

To express the domain in interval notation, we write it as $[-5, \infty)$ . This means that x can be any number greater than or equal to -5.

To find the range of the function, we need to consider the minimum and maximum values that the function can take. The square root function $\sqrt{x}$ always gives non-negative values, and by subtracting3, the minimum value the function can take is -3.

The maximum value of the function is $\infty$ , because as x approaches $\infty$ , the value of $\sqrt{x+5}-3$ also approaches $\infty$ .

So, the range of the function in inequality notation is $y \geq -3$ .