Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find \((\mathbf{a})\) \(\boldsymbol{f} \circ \boldsymbol{g},(\mathbf{b}) \boldsymbol{g} \circ \boldsymbol{f},\) and, if possible, \((\mathbf{c})(\boldsymbol{f} \circ \boldsymbol{g})(\mathbf{0}).\) $$f(x)=3 x+5, \quad g(x)=5-x$$

Step-by-Step Solution

Answer

\(\boldsymbol{f} \circ \boldsymbol{g} = 20 - 3x\), \(\boldsymbol{g} \circ \boldsymbol{f} = -3x\), \((\boldsymbol{f} \circ \boldsymbol{g})(0) = 20.\)

Step 1: Defining composite function \(\boldsymbol{f} \circ \boldsymbol{g}\)

Substitute the expression for \(g(x)\) into \(f(x)\). Thus, the composite function \(f(g(x)) = f(5 - x) = 3(5 - x) + 5\). Simplify this to get \(15 - 3x + 5 = 20 - 3x\).

Step 2: Defining composite function \(\boldsymbol{g} \circ \boldsymbol{f}\)

Now substitute the expression for \(f(x)\) into \(g(x)\). Thus, the composite function \(g(f(x)) = g(3x + 5) = 5 - (3x + 5)\). Simplify this to get \(5 - 3x - 5 = -3x\).

Step 3: Evaluate \((\boldsymbol{f} \circ \boldsymbol{g})(\mathbf{0})\)

Substitute \(x = 0\) into \(f(g(x)) = 20 - 3x\). This gives \(20 - 3(0) = 20\).

Key Concepts

Function Composition

Function composition is like a math version of a nesting doll, where one function 'hides' inside another. When composing two functions, denoted by \( f \circ g \) or \( f(g(x)) \), we essentially insert the entirety of one function into the other. It's critical to follow the order written; \( f \circ g \) is not the same as \( g \circ f \). We apply \( g \) first to an input, and then we apply \( f \) to whatever \( g \) spits out.

Let's make this concrete: imagine \( f(x) \) is your friend giving you a sticker for every dollar you give them, \( f(x) = x \text{ stickers} \), and \( g(x) \) is a machine that converts apples into dollars, \( g(x) = 2x \text{ dollars} \) for every apple. If you feed the machine an apple and take the output dollars to your friend, you're composing \( f \circ g \): for each apple \( x \) you start with, your friend ultimately gives you \( 2x \text{ stickers} \) — that's your composite function in action!

Algebraic Functions

Now, let's dive into the world of algebraic functions, which are a mix-and-match of numbers, variables (like \( x \)), and algebraic operations (like addition and multiplication). They're the bread and butter of high school algebra and offer a playground for examining how different inputs produce different outputs.

For example, if \( f(x) = 3x + 5 \) and \( g(x) = 5 - x \), both are algebraic functions. They're like recipes that tell you what to do with your input \( x \). \( f(x) \) says to triple your \( x \) and then add five, while \( g(x) \) tells you to subtract \( x \) from five. Whenever you encounter functions like these, imagine a machine with gears: you throw in a number, crank the handle by following the 'recipe,' and voilà, a new number comes out.

Evaluating Functions

Evaluating a function is just a fancy term for plugging a number into the function's recipe and seeing what you get. It's like saying, \( f(x) \) is your vending machine, where \( x \) is the amount of money you insert, and the output is the snack you receive.

To evaluate a function such as \( f(x) = 3x + 5 \) when \( x = 2 \), you simply replace every \( x \) with 2 so that you get \( f(2) = 3(2) + 5 = 11 \). The math machine processed your input and gave you the number 11. Evaluating composite functions like \( (f \circ g)(x) \) is a two-step snack grab: \( g(x) \) gives you a number which you then feed into \( f(x) \). It's crucial to keep things in order to avoid mixing up your math snack!

Simplifying Expressions

When faced with a math expression that looks like a jungle of numbers and letters, simplifying is like trimming away the excess to reveal a clearer path. You might combine like terms, use the distributive property, or cancel out terms to get something that's easier to work with.

Picture this: you're sorting through a pile of books and you notice you have 3 copies of 'Harry Potter' and 2 more hiding under 'The Hobbit.' You simplify your 'book expression' by putting them all together to realize you have 5 copies of 'Harry Potter.' In algebra, when faced with an expression like \( 15 - 3x + 5 \), you'd simplify by combining the numbers to get \( 20 - 3x \). It's all about making the expression cleaner and more manageable to work with in your subsequent math adventures.