Problem 27 - Algebra Form and Function

Question

The range of the function \(y=9-(x-2)^{2}\) is \(y \leq 9\). Find the range of the functions. $$ y=\sqrt{9-(x-2)^{2}} $$

Step-by-Step Solution

Answer

Answer: The range of the function \(y=\sqrt{9-(x-2)^{2}}\) is \(0 \leq y \leq 9\).

Step 1: State the given range of the function \(y=9-(x-2)^{2}\)

The given function's range is \(y \leq 9\). That means all of the values of \(y\) are less than or equal to 9.

Step 2: Observe the given function

The given function is \(y=\sqrt{9-(x-2)^{2}}\). Notice that this is the square root of the function we know the range of, \(y=9-(x-2)^{2}\).

Step 3: Analyze the properties of a square root function

A square root function, by definition, returns non-negative values that, when squared, give the argument. In other words, the square root function has a range of \(y \geq 0\). This is an important property to remember when finding the range of the given function.

Step 4: Combine the ranges of the two functions

Since the given function is the square root of the function with range \(y \leq 9\), and we know a square root function always returns non-negative values, we can combine these properties to find the new range. The given function takes the square root of values \(y \leq 9\), and since it cannot return negative values, the lowest value of \(y\) will be 0. Therefore, the range of the given function is \(0 \leq y \leq 9\). So, the range of the function \(y=\sqrt{9-(x-2)^{2}}\) is \(0 \leq y \leq 9\).

Key Concepts

Understanding the Square Root Function

The square root function is a fundamental concept in mathematics. It involves finding a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). The range of a square root function is always non-negative values, meaning it includes zero and positive numbers.
This is because squaring any real number, whether negative or positive, results in a non-negative value.

• If you have a negative inside the square root, it usually doesn't give a real number result.
• The square root of a non-negative number is always non-negative.

So, when considering a function like \(y=\sqrt{9-(x-2)^{2}}\), it will only output non-negative values. This is a key point when determining the range of the function.

Delving into Quadratic Functions

Quadratic functions are polynomials of the form \(ax^2 + bx + c\). The graph of a quadratic function is a parabola. If the leading coefficient \(a\) is positive, the parabola opens upwards, and if \(a\) is negative, it opens downwards.
In the original exercise, the function \(y=9-(x-2)^{2}\) is a specific type of quadratic function that opens downwards. This is because there is a negative sign in front of the squared term.

• The vertex of this parabola is at the point \((2, 9)\).
• The maximum value of \(y\) is 9 at \(x=2\).

Understanding these characteristics helps in visualizing how the range is obtained, which is crucial when dealing with functions involving square roots.

The Role of Non-Negative Values

Non-negative values are numbers that include zero and any positive number, never negative. This concept is essential when working with square root functions, as they don't produce negative results. In the context of the function \(y=\sqrt{9-(x-2)^{2}}\), the output will be constrained to non-negative values since it is derived from a square root.
Let's consider the properties:

• The expression \(9-(x-2)^{2}\) can become zero, leading the square root to be zero as well.
• The values under the square root ensure that \(0 \leq y\), but since \(9-(x-2)^{2}\) is always \(\leq 9\), \(y\) can also be as high as 3, which is the square root of 9.

By blending these constraints, we determine that the range is \(0 \leq y \leq 3\), perfectly integrating the concept of non-negative values in function range determination.