Problem 34 - Algebra and Trigonometry

Question

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=-\sqrt{25-x^{2}} $$

Step-by-Step Solution

Answer

Domain: \([-5, 5]\); Range: \([-5, 0]\).

Step 1: Understand the Function

The function given is \( f(x) = -\sqrt{25-x^2} \). This represents the negative square root of \( 25 - x^2 \). The expression under the square root, \( 25 - x^2 \), needs special attention as it determines where the function is defined.

Step 2: Determine the Graph

Graph the function using a graphing calculator. Note that \( f(x) \) describes the lower hemisphere of a circle centered at the origin with radius 5, which is represented by the equation \( \sqrt{25 - x^2} \), but the negative sign makes the graph reflect downwards.

Step 3: Identify the Domain

The domain of \( f(x) \) is determined by the values of \( x \) for which the expression under the square root is non-negative, i.e., \( 25 - x^2 \geq 0 \). Solving this inequality, we get \( -5 \leq x \leq 5 \). Therefore, the domain of \( f(x) \) is \([-5, 5]\).

Step 4: Identify the Range

The range is determined from the possible values of \( f(x) = -\sqrt{25 - x^2} \). Since the square root function produces values from 0 to 5, the negative sign flips these values to range from -5 to 0. Therefore, the range of the function is \([-5, 0]\).

Key Concepts

Graphing calculator

A graphing calculator is an excellent tool that helps visualize functions and their behaviors. It allows you to graph functions easily by inputting them into the device. This is especially useful when working with more complex functions like the one in this exercise. For the function \( f(x) = -\sqrt{25-x^2} \), using a graphing calculator can simplify the process of understanding the visual representation.
By entering the function into the graphing calculator:

• Ensure you input the function exactly as it appears, including the negative sign and square root.
• The calculator will plot the curve showing the lower half of a circle.
• Observe the x-values (domain) where the function is defined.

Graphing calculators also allow you to zoom in and out for a better look at the graph's details. This can assist in determining both the domain and the range of the function accurately.

Circle equation

The function \( f(x) = -\sqrt{25-x^2} \) is closely related to a circle equation. The expression \( \sqrt{25 - x^2} \) represents part of a circle centered at the origin with a radius of 5. Normally, the equation for a full circle is \( x^2 + y^2 = 25 \). However, in this case, we are dealing with only part of the circle.
The negative sign in front of the square root indicates that we are considering the lower hemisphere:

• This corresponds to the part of the circle below the x-axis.
• Visually, this means the arc part is going downwards (negative y-values).

This understanding helps when using the graphing calculator, as it confirms that the graph should resemble a semi-circle, reflecting downwards due to the negative sign.

Square root function

The square root function, denoted by \( \sqrt{} \), is essential for understanding functions like \( f(x) = -\sqrt{25-x^2} \). The square root itself typically produces non-negative numbers. Here’s how it works in this context:

• The expression under the square root, \( 25-x^2 \), dictates the x-values where the function is valid.
• The values within the square root must be zero or positive (since square root of a negative number isn't real in basic algebra).

For the given function, the square root values would range from 0 to 5, dictated by \( 25-x^2 \geq 0 \). After taking the square root, the negative sign flips the output values, thus transforming them from positive to negative. This creates an output for our function \( f(x) \) that ranges from -5 to 0, establishing the unique behavior of the graph.