let f(x)=sqrt of 121-x^2

a. graph the function
b. state the domain and range of f(x)

y=sqrt[121-x^2]

a. graph the function
b. state the domain and range of f(x)

when x = +11, y = 0
when x = -11, y = 0

when x = 0, y = 11 never gets bigger

If |x| > 11 , y is imaginary

so parabola is upside down with vertex at (0,11) and zeros at +and - 11

It never goes below the y axis
It never goes above y = 11
That defines the range and
x is from -11 to + 11 defining the domain

so the domain is [-11,11)

To graph the function f(x) = √(121 - x^2), we can start by finding points that lie on the graph.

Let's begin by considering different values of x and substituting them into the equation to find the corresponding y-values for the points on the graph.

When x = -10,
f(-10) = √(121 - (-10)^2) = √(121 - 100) = √21 ≈ 4.58

When x = -5,
f(-5) = √(121 - (-5)^2) = √(121 - 25) = √96 ≈ 9.80

When x = 0,
f(0) = √(121 - 0^2) = √121 = 11

When x = 5,
f(5) = √(121 - 5^2) = √(121 - 25) = √96 ≈ 9.80

When x = 10,
f(10) = √(121 - 10^2) = √(121 - 100) = √21 ≈ 4.58

Now that we have some points, we can plot them on a graph and connect them to form the graph.

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-10 -5 0 5 10 x-axis
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Now let's determine the domain and range of f(x).

The domain of f(x) represents all possible values that x can take. In this case, the square root of any negative number is not defined, since imaginary numbers are not considered. Therefore, for f(x) = √(121 - x^2) to be defined, the expression inside the square root must be non-negative. So, we have:

121 - x^2 ≥ 0

Solving for x, we get:

x^2 ≤ 121

Taking the square root of both sides and considering both positive and negative values, we have:

-11 ≤ x ≤ 11

Thus, the domain of f(x) is -11 ≤ x ≤ 11.

The range of f(x) represents all possible values that y can take. Since the square root of a non-negative number is always non-negative, the range of f(x) is equal to or greater than zero. Therefore, the range of f(x) is y ≥ 0.

To graph the function f(x) = √(121 - x^2), we can follow these steps:

Step 1: Determine the domain of the function:
Since it involves a square root, the radicand (121 - x^2) should be greater than or equal to zero. So, we have the inequality:
121 - x^2 ≥ 0

Solving the inequality:
x^2 ≤ 121
Taking the square root of both sides:
-√121 ≤ x ≤ √121
-11 ≤ x ≤ 11

Therefore, the domain of the function is -11 ≤ x ≤ 11.

Step 2: Determine the range of the function:
To find the range, we can analyze the behavior of the square root function. The square root of a non-negative number always produces a non-negative result. So, the range of f(x) is all non-negative real numbers.
Range of f(x): f(x) ≥ 0

Now, let's move on to graphing the function:

a. Graphing the function:
To graph the function, we need to plot points. We can choose various x-values within the domain and then find the associated y-values.

Let's choose some x-values and compute their respective y-values:
x = -11: f(-11) = √(121 - (-11)^2) = √(121 - 121) = √0 = 0
x = -5: f(-5) = √(121 - (-5)^2) = √(121 - 25) = √96 ≈ 9.8
x = 0: f(0) = √(121 - 0^2) = √121 = 11
x = 5: f(5) = √(121 - 5^2) = √(121 - 25) = √96 ≈ 9.8
x = 11: f(11) = √(121 - 11^2) = √(121 - 121) = √0 = 0

Plotting these points (-11, 0), (-5, 9.8), (0, 11), (5, 9.8), (11, 0), we can observe that the function represents a semicircle, centered at (0, 0) with a radius of 11.

b. Domain and Range of f(x):
From our previous analysis, we found that the domain of f(x) is -11 ≤ x ≤ 11, which means all real numbers between -11 and 11 (inclusive).

The range of f(x) is f(x) ≥ 0, which means all non-negative real numbers. So, the range is all numbers greater than or equal to zero.