Problem 2 - Calculus for Biology and Medicine

Question

State the range for the given functions. Graph each function. $$ f(x)=x^{2}, x \in[0,1] $$

Step-by-Step Solution

Answer

The range of the function is \([0, 1]\).

Step 1: Understand the Function

The function given is a quadratic function, specifically a parabola, which is defined as \( f(x) = x^2 \). Since it is defined on the interval \( x \in [0, 1] \), we only need to consider the behavior of the function within this domain.

Step 2: Identify Key Values in the Domain

Within the domain \( [0,1] \), evaluate the function at key points to understand its behavior. Calculate \( f(0) = 0^2 = 0 \) and \( f(1) = 1^2 = 1 \).

Step 3: Determine the Behavior of the Function

Since \( f(x) = x^2 \) is a continuous and increasing function on the interval \([0, 1]\), it reaches every value between \( f(0) = 0 \) and \( f(1) = 1 \) within this domain.

Step 4: State the Range

The range of the function is the set of all possible output values. Since the function increases from \( f(0) = 0 \) to \( f(1) = 1 \), the range is \([0, 1]\).

Step 5: Graph the Function

Graph the function \( f(x) = x^2 \) using the domain \([0, 1]\). The graph begins at (0,0), curves upwards, and ends at (1,1), forming a segment of a parabola. Each point on the curve represents \( (x, x^2) \) for \( x \in [0, 1] \).

Key Concepts

Quadratic Functions

Quadratic functions are an essential part of algebra and calculus. They have the general form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). These functions produce a U-shaped graph known as a parabola. The coefficient \( a \) determines the direction of the parabola: if \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.
To understand a quadratic function, focus on its key features:

• Vertex: The highest or lowest point of the parabola.
• Axis of symmetry: A vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
• Intercepts: Points where the graph crosses the x-axis and y-axis.

For the specific function \( f(x) = x^2 \), which is a simple quadratic function, the value of \( a = 1 \), and both \( b \) and \( c \) are \( 0 \). This means the vertex is at the origin \((0, 0)\), and the parabola is symmetric about the y-axis.

Parabolas

A parabola is a symmetric curve that represents the graph of a quadratic function. It can take different shapes and orientations based on the quadratic's equation. Every parabola has a couple of distinctive characteristics that help to define its shape:

• The direction the parabola opens (upward or downward) is determined by the coefficient \( a \) in the quadratic formula \( ax^2 + bx + c \).
• The vertex is the high or low point depending on the parabola's orientation. In \( f(x) = x^2 \), the vertex is \( (0, 0) \).
• The axis of symmetry is always a vertical line that passes through the vertex, represented by \( x = -\frac{b}{2a} \). For \( f(x) = x^2 \), the axis of symmetry is simply \( x = 0 \).

These properties help in sketching or understanding the parabola's form and behavior within its domain. When the function has restricted domains, like \( x \in [0, 1] \), only a portion of the parabola is considered, leading to specific sections utilized in application or problem-solving.

Graphing Functions

Graphing functions is a useful way to visualize their behavior and identify specific traits like ranges and trends. When graphing quadratic functions such as \( f(x) = x^2 \), we focus on both the algebraic expression and its graphical representation.To graph \( f(x) = x^2 \) over the domain \([0, 1]\), follow these steps:

• Identify the domain and determine critical points. For this function, critical points include \( x = 0 \) and \( x = 1 \) since these are the boundaries of our domain.
• Calculate the function's value at these critical points. Here, \( f(0) = 0 \) and \( f(1) = 1 \).
• Recognize the nature of the function. Since \( f(x) = x^2 \) is continuous and increasing within this domain, it's straightforward to predict the curve's behavior.
• Plot the graph starting from point \( (0, 0) \) up to point \( (1, 1) \), ensuring a smooth upward curve representing a part of a parabola.

Graphing confirms that the range of the function \( f(x) = x^2 \) over the interval \( x \in [0, 1] \) is \([0, 1]\), showing all values between this interval are covered efficiently.