Problem 4 - Intermediate Algebra

Question

For the functions \(f\) and \(g,\) find \(a .(f+g)(x), b .(f-g)(x)\) \(c .(f \cdot g)(x),\) and \(d .\left(\frac{f}{g}\right)(x) .\) $$f(x)=x^{2}-2, g(x)=3 x$$

Step-by-Step Solution

Answer

\((f+g)(x) = x^2 + 3x - 2, (f-g)(x) = x^2 - 3x - 2, (f \cdot g)(x) = 3x^3 - 6x, \left(\frac{f}{g}\right)(x) = \frac{x}{3} - \frac{2}{3x}\).

Step 1: Calculate \((f+g)(x)\)

To find \((f+g)(x)\), we need to add the functions \(f(x)\) and \(g(x)\). So, we have: \[ (f+g)(x) = f(x) + g(x) = x^2 - 2 + 3x \] Thus, \((f+g)(x) = x^2 + 3x - 2\).

Step 2: Calculate \((f-g)(x)\)

To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\). Therefore: \[ (f-g)(x) = f(x) - g(x) = x^2 - 2 - 3x \] This simplifies to \((f-g)(x) = x^2 - 3x - 2\).

Step 3: Calculate \((f \cdot g)(x)\)

For the product of the functions, multiply \(f(x)\) by \(g(x)\): \[ (f \cdot g)(x) = f(x) \cdot g(x) = (x^2 - 2)(3x) \] Distribute to get: \[ (f \cdot g)(x) = 3x^3 - 6x \]

Step 4: Calculate \(\left(\frac{f}{g}\right)(x)\)

For the division of the functions, divide \(f(x)\) by \(g(x)\): \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 2}{3x} \] This can be simplified by dividing each term in the numerator by \(3x\): \[ \frac{x^2}{3x} - \frac{2}{3x} = \frac{x}{3} - \frac{2}{3x} \]

Key Concepts

Understanding Function Addition

Function addition is all about combining two functions to form a new one by adding them. It's similar to adding numbers, but with expressions. When you add two functions like \(f(x)\) and \(g(x)\), you essentially add their outputs for any input \(x\).

For the given functions, \(f(x) = x^2 - 2\) and \(g(x) = 3x\), function addition involves:

• Taking each part separately:
  • the quadratic part \(x^2\),
  • the linear part \(3x\),
  • and the constant \(-2\).
• Summing them: \((f+g)(x) = x^2 + 3x - 2\).

So, this new function returns a value that is the sum of the values of \(f\) and \(g\) for any \(x\).

Exploring Function Subtraction

Function subtraction is similar to function addition but involves subtracting one function from another. In our example, we're using \(f(x) = x^2 - 2\) and \(g(x) = 3x\).

To perform function subtraction (\(f-g)(x)\):

• Start by writing both functions: \(x^2 - 2\) for \(f(x)\) and \(3x\) for \(g(x)\).
• Subtract \(g(x)\) from \(f(x)\):
  • This changes the coefficient of \(3x\) to its negative equivalent, so \((f-g)(x) = x^2 - 3x - 2\).

The resulting function gives the difference in output values of \(f\) and \(g\) for any given input \(x\).

Function Multiplication Explained

Function multiplication involves taking two functions and creating a new function that represents their product. Given our functions, \(f(x) = x^2 - 2\) and \(g(x) = 3x\), multiplying them means:

• Multiply each part of \(f(x)\) by each part of \(g(x)\):
  • \(x^2 \cdot 3x\) results in \(3x^3\),
  • \(-2 \cdot 3x\) results in \(-6x\).
• Combine those products to derive the new function: \((f \cdot g)(x) = 3x^3 - 6x\).

The output of this function gives the multiplied values of \(f\) and \(g\) for any input \(x\). This operation creates a polynomial that is typically of a degree resulting from the sum of the degrees of the two original functions.

Understanding Function Division

Function division involves dividing one function by another, creating a new function that represents this division. For \(f(x) = x^2 - 2\) and \(g(x) = 3x\), to compute function division, the process is as follows:

• Write the division of \(f(x)\) by \(g(x)\):
  • \(\left(\frac{f}{g}\right)(x) = \frac{x^2 - 2}{3x}\).
• Simplify the expression by breaking down each term:
  • Divide \(x^2\) by \(3x\) to get \(\frac{x}{3}\).
  • Divide \(-2\) by \(3x\) to obtain \(-\frac{2}{3x}\).

Thus, you achieve \(\left(\frac{f}{g}\right)(x) = \frac{x}{3} - \frac{2}{3x}\).
This gives a new function where each part of the numerator is divided by the entire denominator, ensuring a proper handling of terms without common factors in the denominator.