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Chapter 4: Problem 108

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

Short Answer

Expert verified
The product \((fg)(x)\) is \(4x^2 - 23x - 35\).

Step by step solution

01

Identify the given functions

The functions provided are: \( f(x) = x - 7 \) \( g(x) = 4x + 5 \)
02

Write down the definition of the product of functions

The product of two functions \( f \) and \( g \) is defined as: \( (fg)(x) = f(x) \cdot g(x) \)
03

Substitute the given functions into the product formula

Substitute \( f(x) = x - 7 \) and \( g(x) = 4x + 5 \) into the product formula: \( (fg)(x) = (x - 7)(4x + 5) \)
04

Apply the distributive property

Use the distributive property to multiply the expressions: \[ (x - 7)(4x + 5) = x \cdot 4x + x \cdot 5 - 7 \cdot 4x - 7 \cdot 5 \]
05

Simplify the resulting expression

Multiply each term and then combine like terms: \[ x \cdot 4x = 4x^2 \] \[ x \cdot 5 = 5x \] \[ -7 \cdot 4x = -28x \] \[ -7 \cdot 5 = -35 \] Combine like terms: \[ 4x^2 + 5x - 28x - 35 = 4x^2 - 23x - 35 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Composition
Function composition is a process of combining two functions so that the output from one function becomes the input of the next. While this concept is different from the product of functions, it鈥檚 useful to understand how these operations differ. When composing functions, if you have two functions, say \(f(x)\) and \(g(x)\), then the composition is written as \((f \bigcirc g)(x)\) or \(f(g(x))\). For example, if \(f(x) = x - 7\) and \(g(x) = 4x + 5\), to find the composition \(f(g(x))\), you substitute \(g(x)\) into \(f(x)\):

  • First, find \(g(x)\) which is \(4x + 5\).
  • Then, substitute this into \(f\): \(f(g(x)) = f(4x+5) = (4x+5) - 7 = 4x - 2\).
This helps you understand how combining functions work sequentially.
Distributive Property
The distributive property is a fundamental principle used to simplify expressions and solve equations. It states that for all real numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) equals \(ab + ac\). When dealing with the product of functions, this property is used to distribute terms across parentheses. Let鈥檚 take an example:

  • Given \((x - 7)(4x + 5)\), we apply the distributive property:
  • Distribute each term in the first parenthesis to each term in the second: \(x \times 4x\), \(x \times 5\), \(-7 \times 4x\), \(-7 \times 5\).
  • Calculate each product: \(4x^2\), \(5x\), \(-28x\), \(-35\).
This gives us a clear path to simplify our expressions.
Simplifying Expressions
Simplifying expressions involves combining like terms and reducing expressions to their simplest form. This not only makes it easier to work with the expressions but also helps in finding accurate solutions quickly. During the product of functions, you'll often get extended expressions that need simplification:

  • After applying the distributive property, the expression from \((x - 7)(4x + 5)\) becomes \(4x^2 + 5x - 28x - 35\).
  • Combine like terms: \(5x - 28x = -23x\).
  • The simplified expression is thus \(4x^2 - 23x - 35\).
Reducing it to this form ensures clarity and correctness.

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