Problem 2 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The range of the function is \([0, 4]\).

Step 1: Understand the Function

The function given is a quadratic function, specifically a parabola opening upwards, since the term with the highest power, \(x^2\), has a positive coefficient. This is a fundamental characteristic of quadratic functions.

Step 2: Identify the Domain

The function is defined for \(x\) within the closed interval \([0, 2]\). This means \(x\) can be any real number between 0 and 2, inclusive.

Step 3: Determine the Behavior of the Function in the Domain

Since \(f(x) = x^2\) is increasing in the interval \([0, 2]\), as \(x\) increases from 0 to 2, the value of \(f(x)\) also increases, beginning at 0 when \(x = 0\) and reaching 4 when \(x = 2\).

Step 4: Find the Range

To find the range, evaluate \(f(x)\) at the boundaries of the domain. \(f(0) = 0\) and \(f(2) = 4\). Since the function is monotonically increasing in the given interval, the range of \(f(x)\) is all values from the minimum \(f(0)\) to the maximum \(f(2)\), hence the range is \([0, 4]\).

Step 5: Graph the Function

Plot the points for \(x = 0\), \(x = 1\), and \(x = 2\) on the Cartesian plane. At \(x = 0\), \(f(x) = 0\); at \(x = 1\), \(f(x) = 1\); at \(x = 2\), \(f(x) = 4\). Since the points form part of a parabola, sketch the curve smoothly through these points, starting at point \((0, 0)\) and ending at point \((2, 4)\).

Key Concepts

Quadratic Functions

Quadratic functions are a specific type of polynomial function where the highest degree of the variable is squared, represented as \( f(x) = ax^2 + bx + c \). In these functions, the graph forms a distinctive curve known as a parabola. A key feature of parabolas is their symmetry, centered around a vertical line called the axis of symmetry. This axis runs through the vertex of the parabola, which is the point where it turns. Quadratics have many important properties:

• The direction the parabola opens (up or down) depends on the coefficient of the \( x^2 \) term (if \( a > 0 \), it opens upwards; if \( a < 0 \), it opens downwards).
• Vertex: This is either the highest or lowest point of the parabola, depending on the direction it opens.
• The solutions to the equation \( f(x) = 0 \) (where the parabola crosses the x-axis) are known as roots or zeroes.

Graphing Functions

Graphing functions involves plotting points derived from input-output pairs and then connecting them to illustrate the overall shape of the function. For quadratic functions like \( f(x) = x^2 \), the process involves following the equation over its domain. To graph a function:

• Start by selecting points within the domain. Here, the domain is the interval \([0, 2]\).
• Calculate the corresponding \( f(x) \) values. For instance, at \( x = 0 \), \( f(x) = 0 \), and at \( x = 2 \), \( f(x) = 4 \).
• Plot these points on a coordinate plane.
• Since quadratic functions graph as parabolas, smoothly connect the plotted points.

Remember to consider the pattern: increasing functions rise as x increases, which is evident as \( f(x) \) rises from 0 to 4 over the domain.

Domain and Range

Understanding the domain and range is crucial when studying functions. The domain of a function is the complete set of possible input values (\( x \) values) that the function can accept without causing errors like division by zero or negative square roots. In our example, the domain is \([0, 2]\). This means only \( x \) values from 0 to 2 are considered.Range, in contrast, refers to the set of possible output values (\( f(x) \)) that the function can produce. For the quadratic \( f(x) = x^2 \) within this domain:

• The smallest output (or minimum value) occurs at \( x = 0 \, (f(x) = 0)\).
• The largest output is at \( x = 2 \, (f(x) = 4)\).
• The function is continuous, and because \( f(x) \) consistently increases, the range includes all values falling between the minimum and maximum. Therefore, the range is \([0, 4]\).

Understanding both concepts helps in properly defining and working with functions.