Problem 1 - Algebra 2

Question

Find \((f+g)(x),(f-g)(x),(f \cdot g)(x),\) and \(\left(\frac{f}{g}\right)\) for each \(f(x)\) and \(g(x)\) $$ \begin{array}{l}{f(x)=3 x+4} \\ {g(x)=5+x}\end{array} $$

Step-by-Step Solution

Answer

\((f+g)(x) = 4x + 9\), \((f-g)(x) = 2x - 1\), \((f \cdot g)(x) = 3x^2 + 19x + 20\), and \(\left(\frac{f}{g}\right)(x) = \frac{3x + 4}{5 + x}\) with \(x \neq -5\).

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

To find \((f+g)(x)\), add the functions \(f(x)\) and \(g(x)\) together.\[\begin{align*}(f+g)(x) &= f(x) + g(x) \&= (3x + 4) + (5 + x) \&= 3x + 4 + 5 + x \&= (3x + x) + (4 + 5) \&= 4x + 9\end{align*}\].

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

To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\). \[\begin{align*}(f-g)(x) &= f(x) - g(x) \&= (3x + 4) - (5 + x) \&= 3x + 4 - 5 - x \&= (3x - x) + (4 - 5) \&= 2x - 1\end{align*}\]

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

To find \((f \cdot g)(x)\), multiply \(f(x)\) and \(g(x)\) together. \[\begin{align*}(f \cdot g)(x) &= (3x + 4) \cdot (5 + x) \&= 3x(5) + 3x(x) + 4(5) + 4(x) \&= 15x + 3x^2 + 20 + 4x \&= 3x^2 + 19x + 20\end{align*}\]

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

To find \(\left(\frac{f}{g}\right)(x)\), divide \(f(x)\) by \(g(x)\). \[\left(\frac{f}{g}\right)(x) = \frac{3x + 4}{5 + x}\]. Make sure to state the restriction that \(x eq -5\), since \(g(x)\) should not equal zero.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to positive integer exponents and coefficients. They're a key part of algebra, forming the basis for much of the mathematical studies. In a polynomial, you might find terms like \(x^2\), \(3x\), or constants like 4. The general form of a polynomial is \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where each \(a_i\) is a coefficient and the largest exponent \(n\) is the degree of the polynomial.
Polynomials are incredibly versatile and can represent a wide range of shapes and patterns. In the given functions \(f(x) = 3x + 4\) and \(g(x) = 5 + x\), both are linear polynomials with a degree of 1. This means each is a straight line. The coefficients in these functions (3 in \(f(x)\) and 1 in \(g(x)\)) determine the slope of these lines, while the constant terms (4 in \(f(x)\) and 5 in \(g(x)\)) provide the y-intercepts.

Function Addition

Function addition involves adding the outputs of two functions at each point in their domain. This means you combine like terms if there are any. With functions \(f(x) = 3x + 4\) and \(g(x) = 5 + x\), function addition is straightforward.
To find \((f+g)(x)\), we simply add \(3x + 4\) and \(5 + x\):

• Combine the \(x\) terms: \(3x + x = 4x\).
• Combine the constants: \(4 + 5 = 9\).

So, \((f+g)(x) = 4x + 9\). This result is a new polynomial function that gives the sum of the values of \(f\) and \(g\) for each input \(x\). This is useful in many scenarios, such as modeling combined effects of different phenomena.

Function Subtraction

Function subtraction is similar to addition but involves taking the difference between their outputs. This requires careful handling to avoid errors in sign. Taking \(f(x) = 3x + 4\) and \(g(x) = 5 + x\), let's find \((f-g)(x)\).
Perform the subtraction by:

• Subtracting the \(x\) terms: \(3x - x = 2x\).
• Subtracting the constants: \(4 - 5 = -1\).

Therefore, \((f-g)(x) = 2x - 1\). This shows how we can determine the rate of change or difference between two varying quantities at each input \(x\). It’s particularly important in calculating differences between data sets or measuring changes over time.

Function Multiplication

Function multiplication involves creating a new function by multiplying the outputs of two functions point-wise. This is often used in more complex scenarios, like scaling effects or interactions between phenomena. For \(f(x) = 3x + 4\) and \(g(x) = 5 + x\), the multiplication process involves the distributive property.
Here's how to calculate \((f \cdot g)(x)\):

• Distribute each term in \(f(x)\) by each term in \(g(x)\):
• \(3x \cdot 5 = 15x\)
• \(3x \cdot x = 3x^2\)
• \(4 \cdot 5 = 20\)
• \(4 \cdot x = 4x\)

After distribution, combining all terms gives: \(3x^2 + 19x + 20\). The resulting polynomial has a degree of 2, indicating it forms a parabolic curve, showcasing that function multiplication can increase polynomial degrees and create more complex shapes.

Function Division

Function division involves dividing the output of one function by that of another. Care must be taken to avoid division by zero, as this makes the function undefined. For our example, \(f(x) = 3x + 4\) and \(g(x) = 5 + x\), we'll find \(\left(\frac{f}{g}\right)(x)\).
This operation is as simple as putting the formulae into a fraction:\[\left(\frac{f}{g}\right)(x) = \frac{3x + 4}{5 + x}\]While constructing such divisions, it's critical to watch out for where the denominator equals zero. Here, \(x = -5\) would make the denominator zero, so \(x eq -5\). Function division might result in a fractional function, also called a rational function, and establishes constraints due to the denominator, which is key in understanding real-world processes like rates or averages.