Solved on Dec 08, 2023

Find the range of the function $f(x) = \sqrt{x+5} - 3$ using graphing technology.

STEP 1

Assumptions
1. The function given is $f(x) = \sqrt{x+5} - 3$ .
2. We are asked to find the range of the function.
3. We will use graphing technology to find the range.
4. The domain of the function $f(x)$ is the set of all $x$ values for which $f(x)$ is defined.

STEP 2

Identify the domain of the function.
For the square root function $\sqrt{x+5}$ , the expression inside the square root must be greater than or equal to zero.
$x + 5 \geq 0$

STEP 3

Solve the inequality to find the domain of $f(x)$ .
$x \geq -5$
This means that the function $f(x)$ is defined for all $x$ values greater than or equal to $-5$ .

STEP 4

Understand the basic shape of the square root function.
The square root function $y = \sqrt{x}$ is a half-parabola that opens upwards, starting from the origin (0,0).

STEP 5

Apply the transformation to the basic square root function.
The function $f(x) = \sqrt{x+5}$ is a horizontal shift of the basic square root function $y = \sqrt{x}$ to the left by 5 units.

STEP 6

Apply the final transformation to the function.
The function $f(x) = \sqrt{x+5} - 3$ is a vertical shift of the function $y = \sqrt{x+5}$ downwards by 3 units.

STEP 7

Determine the lowest point on the graph of $f(x)$ .
The lowest point on the graph of $f(x)$ occurs at the starting point of the square root function, which is when $x = -5$ .

STEP 8

Calculate the value of $f(x)$ at the lowest point.
$f(-5) = \sqrt{-5 + 5} - 3 = \sqrt{0} - 3 = -3$

STEP 9

Determine the behavior of $f(x)$ as $x$ increases.
As $x$ increases, the value of $\sqrt{x+5}$ increases, and thus $f(x)$ increases without bound.

STEP 10

Conclude the range of $f(x)$ .
Since $f(x)$ starts at $-3$ and increases without bound as $x$ increases, the range of $f(x)$ is all real numbers greater than or equal to $-3$ .
$Range\ of\ f(x): y \geq -3$
The range of the function $f(x) = \sqrt{x+5} - 3$ is $y \geq -3$ .

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Find the range of the function $f(x) = \sqrt{x+5} - 3$ using graphing technology.

STEP 1

Assumptions
1. The function given is $f(x) = \sqrt{x+5} - 3$ .
2. We are asked to find the range of the function.
3. We will use graphing technology to find the range.
4. The domain of the function $f(x)$ is the set of all $x$ values for which $f(x)$ is defined.

STEP 2

Identify the domain of the function.
For the square root function $\sqrt{x+5}$ , the expression inside the square root must be greater than or equal to zero.
$x + 5 \geq 0$

STEP 3

Solve the inequality to find the domain of $f(x)$ .
$x \geq -5$
This means that the function $f(x)$ is defined for all $x$ values greater than or equal to $-5$ .

STEP 4

Understand the basic shape of the square root function.
The square root function $y = \sqrt{x}$ is a half-parabola that opens upwards, starting from the origin (0,0).

STEP 5

Apply the transformation to the basic square root function.
The function $f(x) = \sqrt{x+5}$ is a horizontal shift of the basic square root function $y = \sqrt{x}$ to the left by 5 units.

STEP 6

Apply the final transformation to the function.
The function $f(x) = \sqrt{x+5} - 3$ is a vertical shift of the function $y = \sqrt{x+5}$ downwards by 3 units.

STEP 7

Determine the lowest point on the graph of $f(x)$ .
The lowest point on the graph of $f(x)$ occurs at the starting point of the square root function, which is when $x = -5$ .

STEP 8

Calculate the value of $f(x)$ at the lowest point.
$f(-5) = \sqrt{-5 + 5} - 3 = \sqrt{0} - 3 = -3$

STEP 9

Determine the behavior of $f(x)$ as $x$ increases.
As $x$ increases, the value of $\sqrt{x+5}$ increases, and thus $f(x)$ increases without bound.

STEP 10

Conclude the range of $f(x)$ .
Since $f(x)$ starts at $-3$ and increases without bound as $x$ increases, the range of $f(x)$ is all real numbers greater than or equal to $-3$ .
$Range\ of\ f(x): y \geq -3$
The range of the function $f(x) = \sqrt{x+5} - 3$ is $y \geq -3$ .

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Find the range of the function f(x)=x+5−3f(x) = \sqrt{x+5} - 3f(x)=x+5​−3 using graphing technology.

STEP 1

STEP 2

Identify the domain of the function.For the square root function x+5 \sqrt{x+5} x+5​, the expression inside the square root must be greater than or equal to zero.x+5≥0 x + 5 \geq 0 x+5≥0

STEP 3

Solve the inequality to find the domain of f(x) f(x) f(x).x≥−5 x \geq -5 x≥−5This means that the function f(x) f(x) f(x) is defined for all x x x values greater than or equal to −5-5−5.

STEP 4

Understand the basic shape of the square root function.The square root function y=x y = \sqrt{x} y=x​ is a half-parabola that opens upwards, starting from the origin (0,0).

STEP 5

Apply the transformation to the basic square root function.The function f(x)=x+5 f(x) = \sqrt{x+5} f(x)=x+5​ is a horizontal shift of the basic square root function y=x y = \sqrt{x} y=x​ to the left by 5 units.

STEP 6

Apply the final transformation to the function.The function f(x)=x+5−3 f(x) = \sqrt{x+5} - 3 f(x)=x+5​−3 is a vertical shift of the function y=x+5 y = \sqrt{x+5} y=x+5​ downwards by 3 units.

STEP 7

Determine the lowest point on the graph of f(x) f(x) f(x).The lowest point on the graph of f(x) f(x) f(x) occurs at the starting point of the square root function, which is when x=−5 x = -5 x=−5.

STEP 8

Calculate the value of f(x) f(x) f(x) at the lowest point.f(−5)=−5+5−3=0−3=−3 f(-5) = \sqrt{-5 + 5} - 3 = \sqrt{0} - 3 = -3 f(−5)=−5+5​−3=0​−3=−3

STEP 9

Determine the behavior of f(x) f(x) f(x) as x x x increases.As x x x increases, the value of x+5 \sqrt{x+5} x+5​ increases, and thus f(x) f(x) f(x) increases without bound.

STEP 10

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Find the range of the function $f(x) = \sqrt{x+5} - 3$ using graphing technology.

Identify the domain of the function.
For the square root function $\sqrt{x+5}$ , the expression inside the square root must be greater than or equal to zero.
$x + 5 \geq 0$

Solve the inequality to find the domain of $f(x)$ .
$x \geq -5$
This means that the function $f(x)$ is defined for all $x$ values greater than or equal to $-5$ .

Understand the basic shape of the square root function.
The square root function $y = \sqrt{x}$ is a half-parabola that opens upwards, starting from the origin (0,0).

Apply the transformation to the basic square root function.
The function $f(x) = \sqrt{x+5}$ is a horizontal shift of the basic square root function $y = \sqrt{x}$ to the left by 5 units.

Apply the final transformation to the function.
The function $f(x) = \sqrt{x+5} - 3$ is a vertical shift of the function $y = \sqrt{x+5}$ downwards by 3 units.

Determine the lowest point on the graph of $f(x)$ .
The lowest point on the graph of $f(x)$ occurs at the starting point of the square root function, which is when $x = -5$ .

Calculate the value of $f(x)$ at the lowest point.
$f(-5) = \sqrt{-5 + 5} - 3 = \sqrt{0} - 3 = -3$

Determine the behavior of $f(x)$ as $x$ increases.
As $x$ increases, the value of $\sqrt{x+5}$ increases, and thus $f(x)$ increases without bound.