Problem 1 - Calculus for Biology and Medicine

Question

State the range for the given functions. Graph each function. $$ f(x)=x^{2}, x \in \mathbf{R} $$

Step-by-Step Solution

Answer

The range of \(f(x) = x^2\) is \([0, \infty)\).

Step 1: Understand the Function

The given function is a quadratic function, specifically a standard parabola. It is expressed as \( f(x) = x^2 \). Quadratic functions of the form \( x^2 \) open upwards and have a vertex at \((0,0)\). As \(x\) increases or decreases, \(f(x)\) increases.

Step 2: Determine the Vertex

The vertex of this parabola is at \((0,0)\). This is the minimum point on the graph of the function, which tells us that \(f(x) = x^2\) can never be less than 0.

Step 3: Identify the Behavior

Since the parabola opens upwards, as \(x\) moves away from 0 in both the positive and negative directions, \( f(x) \) will increase without bound. This indicates the function's minimum value is at the vertex (\(f(0) = 0\)) and the function has no maximum value.

Step 4: State the Range

The range of \(f(x) = x^2\) includes all values \(f(x)\) can take for real \(x\). Since the minimum \(f(x)\) is 0 and \(f(x)\) goes to infinity, the range is \([0, \,\infty)\).

Step 5: Graph the Function

To graph the function \(f(x) = x^2\), plot points such as \((-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)\) and draw a smooth curve through them, forming a U-shape that opens upwards. The graph reflects that \(f(x)\) is always non-negative and increases as \(|x|\) increases.

Key Concepts

Parabola

A parabola is a U-shaped curve that represents quadratic functions like \(f(x) = x^2\). These are fundamental graphs in mathematics, especially in algebra. A parabola can open upwards or downwards, depending on the sign of the leading coefficient. In the function \(f(x) = x^2\), the parabola opens upwards because the coefficient of \(x^2\) is positive.

When graphed, a parabola is symmetric with respect to its vertical axis of symmetry. This means if you fold the graph along this axis, both halves will overlay perfectly. For \(f(x) = x^2\), the axis of symmetry is the y-axis.

Key characteristics of a parabola include:

• It has a distinct vertex, which is its highest or lowest point, depending on the orientation.
• It is symmetrical around a single vertical line.
• The arms of the parabola extend infinitely in the direction it opens.

The shape and orientation of the parabola provide essential insights into the behavior of the quadratic function.

Vertex

The vertex is a crucial point on a parabola and represents either the minimum or maximum value of the quadratic function, depending on its orientation. For \(f(x) = x^2\), the vertex is a minimum point, located at \((0,0)\). This is because the parabola opens upwards, meaning the arms extend towards higher values along the y-axis.

Let's explore some characteristics that define the vertex:

• In \(f(x) = x^2\), the vertex \((0,0)\) is the lowest point on the graph.
• It marks the point where the direction changes from decreasing to increasing as \(x\) passes through the vertex point.
• The vertex is integral in determining the function's range and provides the smallest function value for a function like \(f(x) = x^2\).

Understanding the vertex helps in solving and graphing quadratic functions, as it is a reference for the direction and shape of the parabola.

Range of a Function

The range of a function defines all the possible output values (\(f(x)\)) that result from plugging in x-values within the function's domain. For the function \(f(x) = x^2\), which is a basic quadratic function, the range is determined by the characteristics of its parabola.

Key aspects of the range for \(f(x) = x^2\) are:

• Since the parabola opens upwards, the smallest output value is at the vertex, \(f(0) = 0\).
• The function increases without bound as \(x\) moves away from zero, heading positive or negative, leading to no maximum value. Hence, the function's range rises to infinity.
• The range is \[ [0, \infty) \], meaning all non-negative real numbers.

Understanding the range provides insights into the limitations and possibilities of a function's outputs. It helps to know what values can be expected when applying the function to real-world scenarios.