Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Find the domain and range of the function. Then evaluate \(f\) at the given \(x\) -value. \(f(x)=4-x^{2}, x=0\)

Step-by-Step Solution

Answer

The domain of the function \(f(x)=4-x^{2}\) is all real numbers \(\mathbb{R}\), the range is \(-\infty < f(x) \leq 4\), and the value of the function at \(x=0\) is \(4\).

Step 1: Finding the Domain

The domain of the function includes all real numbers because there are no restrictions on the values of \(x\) that make the function undefined. Thus, the domain of this function is \(-\infty < x < \infty\) or \(\mathbb{R}\).

Step 2: Finding the Range

A quadratic function of the form \(f(x) = a-x^{2}\) has a maximum value at \(x=0\) and continues to decrease as \(x\) moves away from zero in either direction. The maximum value of this function is \(f(0)=4\). Therefore, the range of the function is \(-\infty < f(x) \leq 4\).

Step 3: Evaluate the Function at Given x-Value

To find the value of the function at a specific point, plug the given \(x\)-value into the function. Here, the \(x\)-value given is \(0\). So, \(f(0)= 4-(0)^{2} = 4\).

Key Concepts

Domain of a Function

Understanding the domain of a function is key when dealing with quadratic functions like \( f(x) = 4 - x^2 \). For most quadratic functions, unless there is a variable in the denominator or under a square root, the domain includes all real numbers. In this specific function, there aren't any divisions by zero or square roots of negative numbers, which means there are no restrictions.
Therefore, the domain of the quadratic function \( f(x) = 4 - x^2 \) is all real numbers. This can be expressed mathematically as \( -\infty < x < \infty \), or simply \( \mathbb{R} \).

• No division by zero issues
• No square roots of negative numbers
• Domain: \( \mathbb{R} \)

No matter the value you choose for \( x \), you will always get a valid function output.

Range of a Function

The range of a quadratic function depends heavily on its structure. For \( f(x) = 4 - x^2 \), the key lies in understanding that this is an "upside down" parabola because of the negative coefficient in front of \( x^2 \).
Since this parabola opens downwards, the function reaches its peak at \( x = 0 \). Evaluating the function at this point gives us \( f(0) = 4 \), the highest point on the curve. As we move away from zero in either direction, \( x^2 \) grows and \( f(x) \) decreases.
Thus, the range is all values from \( -\infty \) to 4, inclusive of 4, which can be expressed as \( -\infty < f(x) \leq 4 \). The function can produce values lower than 4, but never greater than it.

• Maximum value: 4 at \( x = 0 \)
• Function decreases as \( x \) moves away from zero
• Range: \( -\infty < f(x) \leq 4 \)

Function Evaluation

Evaluating a function involves finding the output value given an input \( x \). For the function \( f(x) = 4 - x^2 \), to find \( f(0) \), you simply substitute the \( x \)-value with 0.
Let's do this step-by-step: 1. Start with the function equation: \( f(x) = 4 - x^2 \).2. Replace every \( x \) in the equation with 0: \( f(0) = 4 - 0^2 \).3. Simplify the expression to find \( f(0) = 4 \).This results in 4, meaning that when \( x = 0 \), the function \( f(x) \) evaluates to 4.

• Identify \( x \)-value to evaluate: \( x = 0 \)
• Substitute into the function: \( f(0) = 4 - 0^2 \)
• Simplification leads to \( f(0) = 4 \)

Function evaluation is an essential skill for understanding how functions behave and predict their outputs for different inputs.