Problem 2 - 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. \(\left(\frac{f}{g}\right)(x)\). $$ f(x)=x+4 ; g(x)=5 x-2 $$

Step-by-Step Solution

Answer

a. \(6x + 2\), b. \(-4x + 6\), c. \(5x^2 + 18x - 8\), d. \(\frac{x+4}{5x-2}\)

Step 1: Define what (f+g)(x) means

To find \((f+g)(x)\), we need to add the functions \(f(x)\) and \(g(x)\) together. This means we will add their expressions.

Step 2: Calculate (f+g)(x)

Substituting the given functions, we have:\[ (f+g)(x) = f(x) + g(x) = (x+4) + (5x-2) \] Combine like terms:\[ x + 5x + 4 - 2 = 6x + 2 \]Therefore, \((f+g)(x) = 6x + 2\).

Step 3: Define what (f-g)(x) means

To find \((f-g)(x)\), we need to subtract the function \(g(x)\) from \(f(x)\). This means we will subtract their expressions.

Step 4: Calculate (f-g)(x)

Substituting the given functions, we have:\[ (f-g)(x) = f(x) - g(x) = (x+4) - (5x-2) \] Distribute the negative sign:\[ x + 4 - 5x + 2 = -4x + 6 \]Therefore, \((f-g)(x) = -4x + 6\).

Step 5: Define what (f \cdot g)(x) means

To find \((f \cdot g)(x)\), we need to multiply the functions \(f(x)\) and \(g(x)\). This means we will multiply their expressions.

Step 6: Calculate (f \cdot g)(x)

Substituting the given functions, we have:\[ (f \cdot g)(x) = (x+4)(5x-2) \]Expand the expression using distribution (FOIL method):\[ x \cdot 5x + x \cdot (-2) + 4 \cdot 5x + 4 \cdot (-2) = 5x^2 - 2x + 20x - 8 \]Combine like terms:\[ 5x^2 + 18x - 8 \]Therefore, \((f \cdot g)(x) = 5x^2 + 18x - 8\).

Step 7: Define what \(\left(\frac{f}{g}\right)(x)\) means

To find \(\left(\frac{f}{g}\right)(x)\), we need to divide the function \(f(x)\) by \(g(x)\). This means we will divide their expressions.

Step 8: Calculate \(\left(\frac{f}{g}\right)(x)\)

Substituting the given functions, we have:\[ \left(\frac{f}{g}\right)(x) = \frac{x+4}{5x-2} \]This fraction represents the division of the two functions. There are no further simplifications possible.Therefore, \(\left(\frac{f}{g}\right)(x) = \frac{x+4}{5x-2}\).

Key Concepts

Addition of Functions

When we add two functions, we combine their outputs for each input value. Imagine two recipes you want to combine to make a super recipe, that's basically what adding functions is like!

For the functions given, where \(f(x) = x + 4\) and \(g(x) = 5x - 2\), you simply add their expressions:

• \(f(x) + g(x) = (x + 4) + (5x - 2)\)

Next, perform the addition by combining like terms:

• \(x + 5x = 6x\)
• \(+4 - 2 = 2\)

Thus, the resulting function from addition is \((f+g)(x) = 6x + 2\).

This new function \((f+g)(x)\) gives the overall effect of the two functions acting together for each input \(x\).

Subtraction of Functions

Subtraction of functions is akin to finding out what remains when you take one recipe and remove ingredients of another. Here, we want to find out the effect of one function minus another function for each input.

With \(f(x) = x + 4\) and \(g(x) = 5x - 2\), subtract the functions:

• \(f(x) - g(x) = (x + 4) - (5x - 2)\)

To carry out this operation, distribute the minus sign across the second function:

• \(x + 4 - 5x + 2 = -4x + 6\)

Hence, the resulting function for subtraction is \((f-g)(x) = -4x + 6\).

This new function tells you the result of removing the effects of \(g\) from \(f\) for each \(x\).

Multiplication of Functions

Multiplying functions means taking the output of one function and multiplying it with the output of another. It's like using two recipes to create a whole new extravagant meal!

For \(f(x) = x + 4\) and \(g(x) = 5x - 2\), you multiply their expressions like so:

• \((f \cdot g)(x) = (x + 4)(5x - 2)\)

Use the distributive property, also known as the FOIL method, to expand the expression:

• \(x \cdot 5x = 5x^2\)
• \(x \cdot (-2) = -2x\)
• \(4 \cdot 5x = 20x\)
• \(4 \cdot (-2) = -8\)

Combine the like terms:

• \(5x^2 + 18x - 8\)

Thus, \((f \cdot g)(x) = 5x^2 + 18x - 8\), which gives a whole new dimension of function from multiplying \(f\) and \(g\).

Division of Functions

Dividing one function by another is like determining the ratio of ingredients between two recipes for each value of \(x\).

Given \(f(x) = x + 4\) and \(g(x) = 5x - 2\), perform the function division:

• \(\left(\frac{f}{g}\right)(x) = \frac{x + 4}{5x - 2}\)

This fraction represents the division of the outputs from these functions for every value of \(x\). Keep in mind, division by a function like \(g(x)\) is only defined where \(g(x)eq 0\), so check and exclude any values where \(5x - 2 = 0\).

The result, \(\left(\frac{f}{g}\right)(x) = \frac{x + 4}{5x - 2}\), illustrates the quotient of the two functions, providing insight into how \(f(x)\) relates to \(g(x)\) for each input.