Problem 4 - Calculus for Biology and Medicine

Question

State the range for the given functions. Graph each function. $$ f(x)=x^{2},-\frac{1}{2}

Step-by-Step Solution

Answer

The range of \( f(x) = x^2 \) for \(-\frac{1}{2} < x < \frac{1}{2}\) is \( 0 < f(x) < \frac{1}{4} \).

Step 1: Understand the Function

The function given is \( f(x) = x^2 \). We are tasked with determining the range for this function when the domain is restricted to \(-\frac{1}{2} < x < \frac{1}{2}\). The function \( f(x) = x^2 \) is a standard parabola that opens upwards and is symmetric about the y-axis.

Step 2: Identify the Domain Interval

The domain given is \(-\frac{1}{2} < x < \frac{1}{2}\). This means we only consider the values of \( x \) between \(-\frac{1}{2}\) and \(\frac{1}{2}\), not inclusive of the endpoints.

Step 3: Calculate the Function Values at Boundary of the Domain

Evaluate \( f(x) \) at the endpoints of the domain interval near \(-\frac{1}{2}\) and \(\frac{1}{2}\). This gives us the lowest and highest values that \( f(x) \) can take.- When \( x = -\frac{1}{2} \), \( f(x) = \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \).- When \( x = \frac{1}{2} \), \( f(x) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \).

Step 4: Analyze the Behavior of the Function within the Interval

Within the interval \(-\frac{1}{2} < x < \frac{1}{2}\), \( x^2 \) takes values starting from just above 0 to just below \( \frac{1}{4} \). This is because squaring a number between -0.5 and 0.5 gives a positive number less than 0.25, with the minimum value closer to 0.

Step 5: Determine the Range

Since \( f(x) = x^2 \) is continuous and increasing on both sides towards \( x = 0 \), the range of \( f(x) \) for \(-\frac{1}{2} < x < \frac{1}{2}\) is \( 0 < f(x) < \frac{1}{4} \). The endpoints are not included because \( x \) never actually reaches \(-\frac{1}{2}\) or \( \frac{1}{2} \) in the open interval.

Key Concepts

Parabola

A parabola is a u-shaped curve that can open upwards or downwards. It is the graph of a quadratic function, such as the one given in the problem, \( f(x) = x^2 \). This particular function represents a parabola that opens upwards, meaning it has a minimum point at the vertex. The vertex of \( f(x) = x^2 \) is at the origin (0,0), which is the lowest point on this curve.

Parabolas have certain key features:

• **Vertex**: The turning point of the parabola. For \( f(x) = x^2 \), the vertex is at the origin.
• **Axis of Symmetry**: A vertical line that runs through the vertex, dividing the parabola into two mirror-image halves. For \( f(x) = x^2 \), this line is \( x = 0 \).
• **Direction**: The way the parabola opens. Upwards for positive coefficients of \( x^2 \) and downwards for negative ones.

When tackling problems about parabolas, always identify these parts, as they help to understand how the function behaves over its domain.

Open Interval

An open interval specifies a range of values for a variable that does not include the endpoints. In mathematical notation, an open interval is expressed with parentheses, for example, \(-\frac{1}{2} < x < \frac{1}{2}\) in this problem. It includes all numbers between the two endpoints but excludes those endpoints themselves.

Consider this in the context of the function \( f(x) = x^2 \): the domain of \( x \) is restricted to this open interval, meaning \( x \) can be any number close to \(-\frac{1}{2} \) or \( \frac{1}{2} \), but not the endpoints themselves.

• **Open intervals** allow us to analyze the range of function values without including boundary effects from the endpoints.
• The endpoints \( -\frac{1}{2} \) and \( \frac{1}{2} \) are crucial for estimating the boundaries of possible values \( f(x) \) can attain.
• In calculus, they are also important in determining limits and continuity.

Symmetric Function

In mathematics, a function is considered symmetric if its graph is a mirror image across a specific line. For the function \( f(x) = x^2 \), it is symmetric about the y-axis. This means if you were to fold the graph along the y-axis, both sides would align perfectly.

• **Y-axis Symmetry**: A function with this type of symmetry is called an "even function." For these functions, \( f(x) = f(-x) \) for all x in the domain.
• For example, \( f(0.4) = (0.4)^2 = 0.16 \) and \( f(-0.4) = (-0.4)^2 = 0.16 \). This verifies that \( f(x) \) is symmetric about the y-axis.

This symmetry is beneficial when calculating ranges and determining behavior, as it allows us to consider only part of the graph and extend the results logically. Symmetry simplifies graph analysis and helps visualize function behavior across its domain.