91Ó°ÊÓ

In mathematics, the domain of a function is the set of all possible input values (usually represented by 'x') for which the function is defined. For our function, which is a generalized square root function, it's essential to ensure the expression under the square root sign is non-negative because square roots of negative values are not real numbers.

Given the function:\[f(x) = -\sqrt{25 - x^2}\],

we need the expression inside the square root to be greater than or equal to zero.
Thus, solve the inequality:

\[25 - x^2 \geq 0\].

Rewriting, we get:
\[-5 \leq x \leq 5\]

This means that the domain of the function is all values of \(x\) from \(-5\) to \(5\).

• Always check the expression inside the square root.


• Set up the inequality and solve for \(x\).


• The domain tells us which values \(x\) can take.

The range of a function is the set of all possible output values (usually represented by 'y') which a function can produce. For the function \(f(x) = -\sqrt{25 - x^2}\), let's determine the range step by step.

1. The square root function, \(\sqrt{25 - x^2}\), will always yield non-negative results (i.e., values greater than or equal to zero) because square roots of non-negative numbers are always non-negative.

2. Since our function is \(-\sqrt{25 - x^2}\), we are multiplying these non-negative results by \(-1\).
This results in non-positive values (i.e., values less than or equal to zero).

3. The maximum value of \(\sqrt{25 - x^2}\) is 5, which occurs when \(x = 0\). Thus the minimum value of our given function \(-\sqrt{25 - x^2}\) is \(-5\) at \(x = 0\).

4. The value of \(\sqrt{25 - x^2}\) decreases as \(x\) approaches \(-5\) or 5, reaching zero when \(x = \pm 5\).
At these points, our function \(-\sqrt{25 - x^2}\) equals 0.

Thus, the range of the function is all values from \(-5\) to 0. In interval notation, it's:

\[-5 \leq f(x) \leq 0\].

Graphing functions allows us to see the shape and behavior of a function. Let's graph the function \(f(x) = -\sqrt{25 - x^2}\) by following a few steps.

1. **Identify Key Points**:
Evaluate the function at strategic points within the domain - specifically, at the endpoints and midpoints.

• At \(x = 0\): \(f(0) = -\sqrt{25} = -5\).


• At \(x = 5\): \(f(5) = -\sqrt{0} = 0\).


• At \(x = -5\): \(f(-5) = -\sqrt{0} = 0\).

2. **Plot the Points**:
Plot the points \((0, -5)\), \((5, 0)\), and \((-5, 0)\).

3. **Draw the Curve**:
Draw a smooth curve connecting these points. The shape resembles an upside-down semicircle opening downward, since it covers the range from \(-5\) to 0 and exists over the domain \([-5, 5]\).

• Always check key values in the domain.


• Plot these points on the graph accurately.


• Draw smooth curves to connect the points.

By understanding key points and plotting them, we can accurately visualize the behavior and characteristics of the function.