Problem 8 - Beginning and Intermediate Algebra

Question

For each pair of functions, find a) \((f g)(x)\) and \(b\) ) \((f g)(-3)\). $$f(x)=4 x+7, g(x)=x-5$$

Step-by-Step Solution

Answer

a) \((f g)(x) = 4x^2 - 13x - 35\) b) \((f g)(-3) = 40\)

Step 1: Find the product function \((f g)(x)\)

To find the product function \((f g)(x)\), we need to multiply the expressions of the given functions f(x) and g(x). The given functions are: \(f(x) = 4x + 7\) and \(g(x) = x - 5\) Now, multiply these expressions: \((f g)(x) = (4x + 7)(x - 5)\)

Step 2: Simplify the expression for \((f g)(x)\)

To simplify the expression for the product function \((f g)(x)\), we will use the distributive property to expand the expression: \((f g)(x)= (4x + 7)(x - 5) = 4x(x - 5) + 7(x - 5)\) Next, multiply the terms: \((f g)(x) = 4x^2 - 20x + 7x - 35\) Now, combine the like terms: \((f g)(x) = 4x^2 - 13x - 35\) So the product function \((f g)(x) = 4x^2 - 13x - 35\)

Step 3: Find the value of \((f g)(-3)\)

Now that we have the product function \((f g)(x) = 4x^2 - 13x - 35\), we can find its value at x = -3: \((f g)(-3) = 4(-3)^2 - 13(-3) - 35\) Simplify the expression: \((f g)(-3) = 4(9) + 39 - 35 = 36 + 39 - 35\) Finally, add the numbers: \((f g)(-3) = 40\) So, \((f g)(-3) = 40\).

Key Concepts

Function Operations

When we talk about function operations, we're referring to ways to combine functions to create new ones. Functions can be combined using various operations like addition, subtraction, multiplication, and division. In this exercise, we focus on multiplication. The notation \((f \cdot g)(x)\) means that we're multiplying the outputs of two functions \(f(x)\) and \(g(x)\) together, resulting in a new function. Here’s how it works:

• Addition and Subtraction: These operations combine functions by adding or subtracting their outputs. For instance, \((f+g)(x) = f(x) + g(x)\).
• Multiplication: Multiply the outputs to form \( (f \cdot g)(x) = f(x) \times g(x) \).
• Division: When dividing, make sure \(g(x)\) isn't zero, noted as \((\frac{f}{g})(x) = \frac{f(x)}{g(x)}\).

Using operation like this allows us to explore new mathematical relationships and set the stage for more advanced function concepts.

Polynomial Functions

Polynomial functions are like the versatile tools of algebra. They're made of variables (usually x) raised to different powers and then added or subtracted together. This power can be a positive integer, and each term in the polynomial is made up of a coefficient and a variable raised to an exponent.

• Terms: Each piece of a polynomial, such as \(4x^2\), \(-13x\), or \(-35\) in our example, is a term.
• Degree: The degree of a polynomial is the highest power of its variable. In \(4x^2 - 13x - 35\), it's 2.
• Standard Form: A polynomial is in standard form when its terms are written in descending order of their degrees, like \(4x^2 - 13x - 35\).

In the step-by-step solution, we simplified the expression by combining like terms to arrive at a polynomial function \((f g)(x)\). Such exercises help sharpen our skills in manipulating and understanding these foundational algebraic structures.

Distributive Property

The distributive property is a vital tool in algebra, allowing us to simplify expressions. It states that multiplying a number by a sum is the same as multiplying each addend separately and then adding their products. The property is generally noted as \(a(b + c) = ab + ac\).

In the provided solution, we see the distributive property in action when expanding \((4x + 7)(x - 5)\). Here’s how it works step by step:

• First distribute 4x: We multiply \(4x\) by each term inside the second bracket. So, \(4x(x) = 4x^2\) and \(4x(-5) = -20x\).
• Next, distribute 7: Multiply \(7\) by the terms inside the same brackets: \(7(x) = 7x\) and \(7(-5) = -35\).
• Combine like terms: Finally, simplify the expression \(4x^2 - 20x + 7x - 35\) to \(4x^2 - 13x - 35\).

Mastering the distributive property is crucial as it lays the groundwork for more complex algebraic manipulations. It's a key strategy in algebra for simplifying expressions, solving equations, and proving identities.