Problem 20 - Algebra and Trigonometry

Question

Draw the graphs of \(f, g,\) and \(f+g\) on a common screen to illustrate graphical addition. $$ f(x)=\sqrt[4]{1-x}, g(x)=\sqrt{1-\frac{x^{2}}{9}} $$

Step-by-Step Solution

Answer

Graph the functions \(f(x)\), \(g(x)\), and \(f(x)+g(x)\) over their valid domains, then observe the changes.

Step 1: Understanding the Functions

We have two functions: \(f(x) = \sqrt[4]{1-x}\) and \(g(x) = \sqrt{1-\frac{x^2}{9}}\). The function \(f(x)\) is defined for \(1-x \geq 0\), which means the domain is \(x \leq 1\). The function \(g(x)\) describes the upper semicircle of the ellipse given by \(x^2/9 + y^2 = 1\), hence its domain is \(-3 \leq x \leq 3\).

Step 2: Finding the Domain for Addition

The domain of \(f(x) + g(x)\) will be the intersection of the domains of \(f\) and \(g\), which is \(-3 \leq x \leq 1\).

Step 3: Graphing the Function \(f(x)\)

Plot the graph of \(f(x) = \sqrt[4]{1-x}\) over the domain \(-3 \leq x \leq 1\). This graph will start from \(f(-3)\) which is imaginary and real values begin only \(x=1\) going downwards as \(x\) approaches \(-3\).

Step 4: Graphing the Function \(g(x)\)

Plot the graph of \(g(x) = \sqrt{1-\frac{x^2}{9}}\). It will trace the upper half of an ellipse with a horizontal range from \(-3\) to \(3\).

Step 5: Graphing \(f(x) + g(x)\)

Finally, plot the graph of \(f(x) + g(x)\) which adds the values from \(f(x)\) and \(g(x)\) across their common domain, \(-3 \leq x \leq 1\). For any \(x\), the value is \(\sqrt[4]{1-x} + \sqrt{1-\frac{x^2}{9}}\).

Step 6: Analyzing the Graphs

Compare the individual graphs of \(f(x)\) and \(g(x)\) with their sum. Notice the shape changes indicating the effect of graphical addition.

Key Concepts

Domain of Functions

When we talk about the domain of a function, we mean the set of all possible input values (usually represented as "x") that the function can accept without causing any mathematical errors. The domain is very important because it helps us understand the limits within which the function behaves normally and can be graphed.
Let's examine two functions:

• The function \(f(x) = \sqrt[4]{1-x}\) has a domain defined by the condition that everything inside the root should be non-negative. So, \(1-x \geq 0\), leading to \(x \leq 1\). This means \(f(x)\) is only defined and will produce real numbers when \(x\) is less than or equal to 1.
• The function \(g(x) = \sqrt{1-\frac{x^2}{9}}\) represents the upper half of an ellipse. Its domain comes from ensuring the expression inside the square root remains non-negative, giving us \(x^2/9 \leq 1\), which simplifies to \(-3 \leq x \leq 3\).

Understanding the domains of these functions is crucial for graphing them accurately. By recognizing these intervals, we prepare for correctly displaying the functions on a plot.

Plotting Graphs

Graphing functions involves illustrating their behavior on a coordinate plane. It is a visual tool that helps comprehend mathematical relationships. Before starting, it's essential to understand the domain, as it shows us where the function is happy and healthy.
For example:

• The graph of \(f(x) = \sqrt[4]{1-x}\) will appear only on the left side of the coordinate plane up until \(x=1\). On plotting, you will notice the graph starts from 1 and declines as you move left, within the allowable domain. Imaginary numbers begin to affect if you move right of \(x=1\).
• The plot \(g(x) = \sqrt{1-\frac{x^2}{9}}\), on the other hand, draws out the top portion of an ellipse, existing between \(-3\) and \(3\). As you plot this, you'll see a smooth curve intercepting the x-axis at \(-3, 0\) and \(3, 0\).

Correct plotting provides an accurate visual representation, allowing easy identification of characteristics like intercepts and asymptotic behavior.

Intersection of Domains

The intersection of domains occurs when we want to perform operations involving multiple functions, and it involves finding the "common ground" between them. This concept is important because it determines the new domain where combined functions like \(f(x) + g(x)\) behave well together.
For the functions \(f(x) = \sqrt[4]{1-x}\) and \(g(x) = \sqrt{1-\frac{x^2}{9}}\), their intersection means finding x-values where both functions are defined.
Given the domains:

• \(f(x)\) is valid where \(x \leq 1\)
• \(g(x)\) exists where \(-3 \leq x \leq 3\)

Their intersection is \(-3 \leq x \leq 1\). This implies that the domain of the function \(f(x) + g(x)\) is limited to this overlap. Essentially, only within this common interval are both original functions well-behaved enough to be summed graphically. This intersection extends our understanding and respects function constraints.