Question

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=\sqrt{25-x^{2}}\) \(x=0\)

Short answer

The domain of \(f(x)=\sqrt{25-x^{2}}\) is \([-5,5]\), the range is \([0,5]\), and \(f(0)=5\).

Step-by-step solution

Determine the Domain

To find the domain of the function \(f(x) = \sqrt{25 - x^2}\), find the values of x that make the expression under the square root sign non-negative. To do this, set \(25 - x^2 \geq 0\) which simplifies to \(-x^2 \geq -25\). Divide both sides by -1 to get \(x^2 \leq 25\). Thus x is between -5 and 5, inclusive; so the domain is \([-5,5]\).

Determine the Range

Since the expression under the square root is always non-negative and we are dealing with real numbers, the function \(f(x) = \sqrt{25 - x^2}\) will output values between 0 and 5 (inclusive). Therefore, the range of the function is \([0,5]\).

Evaluate the Function at the Given Point

Evaluating the function at \(x = 0\) implies substituting \(x = 0\) into the function. Then we have \(f(0) = \sqrt{25 - 0^2}\) which equals 5.

Key concepts

Evaluating Functions

When we talk about evaluating functions, we refer to the process of finding the output of a function for a particular input. Imagine you have a vending machine; when you select a button associated with a snack (input), the machine gives you the chosen snack (output). Likewise, when you have a function defined by an expression, such as \(f(x) = \sqrt{25 - x^2}\), evaluating the function for a specific value involves replacing the variable \(x\) with the given number and calculating the result.

To evaluate \(f\) at \(x = 0\), you simply substitute \(0\) for \(x\) in the function and solve for \(f(0)\). So, \(f(0) = \sqrt{25 - 0^2} = \sqrt{25} = 5\). This is the function's output when \(x\) is zero. Understanding how to evaluate functions is vital in graphing them, solving equations, and analyzing their behavior.

Square Root Functions

Square root functions are a type of radical function characterized by the square root sign. In a function like \(f(x) = \sqrt{25 - x^2}\), the square root determines the shape of its graph and defines its domain and range. For the square root of a number to be a real number (since we typically focus on real numbers in algebra), the value inside the square root (the radicand) must be non-negative.

Characteristics of Square Root Functions

• The domain must ensure the radicand is \(\geq 0\).
• The range is typically non-negative because square roots yield positive results or zero.
• The graph of a square root function usually takes on a 'half-parabola' shape.
• They are used in various applications, including geometry and physics.

Understanding these functions helps students solve equations that model real-world scenarios, such as projectile motion.

Function Notation

Function notation, using symbols like \(f(x)\), is a concise way to represent functions. It tells us that \(f\) is a function of \(x\). This notation is practical for showing operations performed on the input \(x\), and for distinguishing between different functions, like \(f(x)\), \(g(x)\), or \(h(x)\).

In our example, \(f(x) = \sqrt{25 - x^2}\) suggests that the function \(f\) will give you the positive square root of \(25 - x^2\) for any \(x\) within the defined domain. Function notation is essential for clearly communicating mathematical concepts and for solving complex problems involving multiple functions.

Inequalities

Inequalities are mathematical expressions that show the relative size or order of two values. They are essential in defining the domain and range of a function. In the case of our square root function, \(f(x) = \sqrt{25 - x^2}\), we use an inequality to set a condition \(25 - x^2 \geq 0\) to find the domain.

Steps to Solve Inequalities

• Write the inequality that must be satisfied by the variable.
• Solve the inequality as you would an equation, remembering to reverse the inequality sign when multiplying or dividing by a negative.

By mastering inequalities, students equip themselves with the tools to tackle complex problems involving range and domain, as well as real-life situations that require optimization and constraint solutions.