Problem 5 - Algebra A Combined Function

Question

For the functions \(f\) and \(g\), find a. \((f+g)(x)\), b. \((f-g)(x), c .(f \cdot g)(x)\), and d. \(\left(\frac{f}{g}\right)(x)\). $$ f(x)=\sqrt{x} ; g(x)=x+5 $$

Step-by-Step Solution

Answer

(f+g)(x) = \sqrt{x} + x + 5; (f-g)(x) = \sqrt{x} - x - 5; (f \cdot g)(x) = x\sqrt{x} + 5\sqrt{x}; \left(\frac{f}{g}\right)(x) = \frac{\sqrt{x}}{x + 5}.

Step 1: Understanding the Functions

We are given two functions: \( f(x) = \sqrt{x} \) and \( g(x) = x + 5 \). The task is to perform basic operations on these functions and find the resulting expressions.

Step 2: Finding \((f+g)(x)\)

To add two functions, just add their expressions. \[ (f+g)(x) = f(x) + g(x) = \sqrt{x} + (x + 5) = \sqrt{x} + x + 5 \]

Step 3: Finding \((f-g)(x)\)

To subtract two functions, subtract their expressions. \[ (f-g)(x) = f(x) - g(x) = \sqrt{x} - (x + 5) = \sqrt{x} - x - 5 \]

Step 4: Finding \((f \cdot g)(x)\)

To multiply two functions, multiply their expressions. \[ (f \cdot g)(x) = f(x) \cdot g(x) = \sqrt{x} \cdot (x + 5) = \sqrt{x} \cdot x + \sqrt{x} \cdot 5 = x\sqrt{x} + 5\sqrt{x} \]

Step 5: Finding \(\left(\frac{f}{g}\right)(x)\)

To divide two functions, divide their expressions. \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{x}}{x + 5} \] Ensure \(x + 5 eq 0\), therefore, \(x eq -5\) to avoid division by zero.

Key Concepts

Function Addition

When you are adding two functions, it simply means you take the output values of both functions and add them together. For the functions \( f(x) = \sqrt{x} \) and \( g(x) = x + 5 \), the sum is straightforward:

• First, evaluate \( f(x) \) and \( g(x) \).
• Then, add the results.

In mathematical terms:\[(f+g)(x) = f(x) + g(x) = \sqrt{x} + (x + 5) = \sqrt{x} + x + 5\]This notation \((f+g)(x)\) represents the sum of the functions evaluated at each \( x \). It combines the expression results into a single simplified function expression.

Function Subtraction

Subtraction of functions involves taking the output of one function and subtracting the output of another function. It's as simple as performing subtraction on their expressions. For our functions \( f(x) = \sqrt{x} \) and \( g(x) = x + 5 \), follow these steps:

• Evaluate each function individually.
• Subtract \( g(x) \) from \( f(x) \).

The resulting function expression is:\[(f-g)(x) = f(x) - g(x) = \sqrt{x} - (x + 5) = \sqrt{x} - x - 5\]Subtraction requires careful attention to parentheses, as they ensure each part of \( g(x) \) is correctly subtracted from \( f(x) \).

Function Multiplication

Multiplying functions requires multiplying the results of two expressions for each \( x \). For \( f(x) = \sqrt{x} \) and \( g(x) = x + 5 \), this involves the product of both functions:

• Multiply \( f(x) \) by \( g(x) \).
• Distribute \( \sqrt{x} \) over the terms in \( g(x) \).

Mathematically, it works as follows:\[(f \cdot g)(x) = f(x) \cdot g(x) = \sqrt{x} \cdot (x + 5)\]Breaking this down:\[= x \sqrt{x} + 5 \sqrt{x}\]Multiplication involves expanding and simplifying expressions, keeping track of terms carefully.

Function Division

Function division divides the results of one function by another. With \( f(x) = \sqrt{x} \) and \( g(x) = x + 5 \), division follows these steps:

• Evaluate both functions.
• Place \( f(x) \) as the numerator and \( g(x) \) as the denominator.
• Ensure the denominator is non-zero.

Mathematically expressed, it looks like this:\[\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{x}}{x + 5}\]It's crucial to note that division is only defined for \( x + 5 eq 0 \). This means \( x eq -5 \) to avoid division by zero. Always check the expression in the denominator to ensure it doesn't simplify to zero.