Problem 56 - Algebra 2

Question

Find \((f+g)(x),(f-g)(x),(f \cdot g)(x),\) and \(\left(\frac{f}{g}\right)(x)\) for each \(f(x)\) and \(g(x)\) $$ \begin{array}{l}{f(x)=x+5} \\ {g(x)=x-3}\end{array} $$

Step-by-Step Solution

Answer

(f+g)(x) = 2x + 2, (f-g)(x) = 8, (f\cdot g)(x) = x^2 + 2x - 15, \left(\frac{f}{g}\right)(x) = \frac{x+5}{x-3} (x \neq 3).

Step 1: Understand Function Addition

To find \((f+g)(x)\), we need to add the functions \(f(x)\) and \(g(x)\). Our functions are \(f(x) = x + 5\) and \(g(x) = x - 3\). To add them, we combine like terms.

Step 2: Perform Function Addition

Substitute \(f(x)\) and \(g(x)\) into \((f+g)(x)\):\[(f+g)(x) = (x + 5) + (x - 3) = x + x + 5 - 3 = 2x + 2\].

Step 3: Understand Function Subtraction

To find \((f-g)(x)\), we subtract the function \(g(x)\) from \(f(x)\). This involves performing the operation \(f(x) - g(x)\), where \(f(x) = x + 5\) and \(g(x) = x - 3\).

Step 4: Perform Function Subtraction

Substitute \(f(x)\) and \(g(x)\) into \((f-g)(x)\):\[(f-g)(x) = (x + 5) - (x - 3) = x + 5 - x + 3 = 8\].

Step 5: Understand Function Multiplication

To find \((f \cdot g)(x)\), multiply \(f(x)\) by \(g(x)\). It involves expanding the expression \((x+5)(x-3)\).

Step 6: Perform Function Multiplication

Apply the distributive property to find \((f \cdot g)(x)\):\[(f \cdot g)(x) = (x+5)(x-3) = x^2 - 3x + 5x - 15 = x^2 + 2x - 15\].

Step 7: Understand Function Division

To find \(\left(\frac{f}{g}\right)(x)\), divide \(f(x)\) by \(g(x)\). We need to express this as a single fraction form.

Step 8: Perform Function Division

Write the division of \(f(x)\) by \(g(x)\):\[\left(\frac{f}{g}\right)(x) = \frac{x+5}{x-3}\]. Watch for division by zero; this expression is undefined when \(x = 3\).

Key Concepts

Understanding Function Addition

When it comes to adding functions, the idea is quite similar to adding numbers or objects. Each of the components of the functions needs to be added respectively. In this case, we have two functions, \(f(x) = x+5\) and \(g(x) = x-3\), and we want to find \((f + g)(x)\). To do this, simply add them together.

• Add the linear terms: \(x + x = 2x\).
• Add the constant terms: \(5 - 3 = 2\).

Therefore, the resulting function after addition is \((f + g)(x) = 2x + 2\). It's that simple! Just add the corresponding parts together—the coefficients of \(x\) and the constants.
Function addition is handy when dealing with problems requiring multiple changes to the same variable, like algebraic manipulations or dynamically changing systems.

Understanding Function Subtraction

Subtraction of functions follows a similar principle to addition but involves removing components rather than combining them. Given our functions \(f(x) = x+5\) and \(g(x) = x-3\), to find \((f-g)(x)\), you subtract every part of \(g(x)\) from the corresponding part of \(f(x)\).

• Cancel the linear terms: \(x - x = 0\).
• Subtract the constant terms: \(5 - (-3) = 5 + 3 = 8\).

So, \((f-g)(x) = 8\). This expression simplifies down to a constant when the linear terms cancel out. Function subtraction might seem straightforward, but it's surprisingly powerful in evaluating differences between processes or entities.

Understanding Function Multiplication

Function multiplication involves multiplying each component of one function with every component of the other function. This is a bit more intricate, and for functions \(f(x) = x+5\) and \(g(x) = x-3\), this involves expanding the expression \((x+5)(x-3)\).

• Apply the distributive property using the terms of the polynomial:
• \(x \cdot x = x^2\)
• \(x \cdot -3 = -3x\)
• \(5 \cdot x = 5x\)
• \(5 \cdot -3 = -15\)

Combine the like terms: \(x^2 - 3x + 5x - 15\), which results in \((f \cdot g)(x) = x^2 + 2x - 15\). Function multiplication is widely used in situations requiring analysis of varying factors, such as in physics and engineering problems.

Understanding Function Division

Division of functions involves placing one function over another in fraction form. In our scenario, this will be \(\left(\frac{f}{g}\right)(x) = \frac{x+5}{x-3}\). This indicates dividing the function \(f(x)\) by \(g(x)\).A special consideration here involves ensuring the denominator isn't zero, as division by zero is undefined. For our functions, \(\left(\frac{f}{g}\right)(x)\) becomes undefined when \(g(x) = x - 3 = 0\), which occurs at \(x=3\).Therefore, you need to include the condition that \(x eq 3\) when stating your solution. Function division is critical in fields like calculus, where it helps in deriving rates of change or analyzing quotient relationships.