Problem 7 - Precalculus

Question

Use the pair of functions \(f\) and \(g\) to find the following values if they exist. \- \((f+g)(2)\) \- \((f g)\left(\frac{1}{2}\right)\) -\((f-g)(-1)\) -\(\left(\frac{f}{g}\right)(0)\) -\((g-f)(1)\) -\(\left(\frac{g}{f}\right)(-2)\) $$ f(x)=2 x \text { and } g(x)=\frac{1}{2 x+1} $$

Step-by-Step Solution

Answer

(f+g)(2)=4.2, (fg)(1/2)=0.5, (f-g)(-1)=-3, (f/g)(0)=0, (g-f)(1)=-5/3, (g/f)(-2)=-1/12.

Step 1: Find (f+g)(x)

To find \[ (f+g)(x) = f(x) + g(x) \] we need to compute: \[ (f+g)(x) = 2x + \frac{1}{2x+1} \] Now, plug in the value 2 to find \[ (f+g)(2) = 2(2) + \frac{1}{2(2)+1} = 4 + \frac{1}{5} = 4.2 \]

Step 2: Find (fg)(x)

The function \[ (fg)(x) = f(x) \cdot g(x) \] can be written as: \[ (fg)(x) = 2x \cdot \frac{1}{2x+1} = \frac{2x}{2x+1} \] Substitute \( x = \frac{1}{2} \):\[ (fg)\left(\frac{1}{2}\right) = \frac{2\left(\frac{1}{2}\right)}{2\left(\frac{1}{2}\right)+1} = \frac{1}{2} \]

Step 3: Find (f-g)(x)

The function \[ (f-g)(x) = f(x) - g(x) \] can be expressed as: \[ (f-g)(x) = 2x - \frac{1}{2x+1} \] Now, we compute it for \( x = -1 \):\[ (f-g)(-1) = 2(-1) - \frac{1}{2(-1)+1} = -2 + 1 = -3 \]

Step 4: Find (f/g)(x)

We need to calculate \[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \]Substituting, we have:\[ \frac{f(x)}{g(x)} = \frac{2x}{\frac{1}{2x+1}} = 2x(2x+1) = 4x^2 + 2x \]Now substitute \( x = 0 \):\[ \left(\frac{f}{g}\right)(0) = 4(0)^2 + 2(0) = 0 \]

Step 5: Find (g-f)(x)

For the expression \[ (g-f)(x) = g(x) - f(x) \]we can write:\[ (g-f)(x) = \frac{1}{2x+1} - 2x \] Substitute \( x = 1 \):\[ (g-f)(1) = \frac{1}{2(1)+1} - 2(1) = \frac{1}{3} - 2 = -\frac{5}{3} \]

Step 6: Find (g/f)(x)

The function \[ \left(\frac{g}{f}\right)(x) = \frac{g(x)}{f(x)} \]can be formulated as:\[ \frac{g(x)}{f(x)} = \frac{\frac{1}{2x+1}}{2x} = \frac{1}{2x(2x+1)} \]Substituting \( x = -2 \):\[ \left(\frac{g}{f}\right)(-2) = \frac{1}{2(-2)(2(-2)+1)} = \frac{1}{-16 + 4} = -\frac{1}{12} \]

Key Concepts

Function Addition

Function addition involves combining two functions to create a third function, known as the sum function. When you add functions, you're essentially adding their outputs or values at specific points. For example, if you have two functions, say \(f(x)\) and \(g(x)\), their sum is expressed as:

• \((f+g)(x) = f(x) + g(x)\)

This means you need to evaluate the sum of the two functions at the same x-value. In the context of the problem, for \(x = 2\), we computed:

• \((f + g)(2) = 2(2) + \frac{1}{2 \cdot 2 + 1} = 4 + \frac{1}{5} = 4.2\)

When performing function addition, remember to simplify the expression, especially when dealing with fractions and whole numbers, to make calculations easier to manage.

Function Multiplication

Multiplying functions involves creating a new function called the product function by combining two existing functions. The process involves multiplying the outputs from each function at the same x-value. If you have two functions \(f(x)\) and \(g(x)\), their product is:

• \((f \cdot g)(x) = f(x) \cdot g(x)\)

For the given exercise, we demonstrated this by evaluating the product of \(f(x)\) and \(g(x)\) at \(x = \frac{1}{2}\):

• \((f g)\left(\frac{1}{2}\right) = \frac{2 \cdot \frac{1}{2}}{2 \cdot \frac{1}{2} + 1} = \frac{1}{2}\)

The steps involved require careful multiplication of terms and simplification, especially when dealing with fractions, to get the correct outcome.

Function Subtraction

Subtracting functions involves finding the difference between the outputs of two functions for the same input value. This operation gives us the difference function. For functions \(f(x)\) and \(g(x)\), the subtraction is:

• \((f - g)(x) = f(x) - g(x)\)

In the problem, to calculate \( (f-g)(-1) \), we substitute \(-1\) into our function expressions:

• \((f - g)(-1) = 2(-1) - \frac{1}{2(-1) + 1}\)
• Which simplifies to \(-2 + 1 = -3\)

Function subtraction can be tricky when it involves fractions. Careful attention to the signs (positive and negative) and simplification helps avoid errors.

Function Division

Dividing functions results in a quotient function. This involves dividing the output of one function by the output of another at the same x-value. If you have the functions \(f(x)\) and \(g(x)\), their quotient is given by:

• \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\)

For the exercise, we evaluated this at \(x = 0\):

• \(\left(\frac{f}{g}\right)(0) = 2(0)(2(0) + 1) = 0\)

Function division requires attention to the denominators, as dividing by zero is undefined and should be avoided. Always simplify the division expression to find a clear result.