Problem 28 - Beginning and Intermediate Algebra

Question

Let \(f(x)=-x^{2}+10 x+4\) and \(g(x)=x+1 .\) Find a) \((g \circ f)(x)\) b) \(\quad(f \circ g)(x)\) c) \((f \circ g)(-2)\)

Step-by-Step Solution

Answer

We expand this as follows: \[ (g \circ f)(x) = g(-x^2 + 10x + 4) = (-x^2 + 10x + 4) + 1. \] #tag_title#Step 2: Simplify (g ○ f)(x)#tag_content#Now we simplify the expression: \[ (g \circ f)(x) = -x^2 + 10x + 4 + 1 = -x^2 + 10x + 5. \] #tag_title#Step 3: Expand (f ○ g)(x)#tag_content#Now, we need to find the second composition, \((f \circ g)(x)\), which means we need to plug the function \(g(x)\) into the function \(f(x)\). This is shown as \[ (f \circ g)(x) = f(g(x)). \] We expand this as follows: \[ (f \circ g)(x) = f(x+1) = -(x+1)^2 + 10(x+1) + 4. \] #tag_title#Step 4: Simplify (f ○ g)(x)#tag_content#Now we simplify the expression: \[ (f \circ g)(x) = - (x^2 + 2x + 1) + 10x + 10 + 4 = -x^2 + 10x + 13. \] #tag_title#Step 5: Compute (f ○ g)(-2)#tag_content#Now we need to find the value of \((f \circ g)(-2)\), which is when \(x = -2\). \[ (f \circ g)(-2) = -(-2)^2 + 10(-2) + 13 = - (-4) - 20 + 13 = -3. \] #tag_title#Answer#tag_content#The compositions are: a) \((g \circ f)(x) = -x^2 + 10x + 5\), b) \((f \circ g)(x) = -x^2 + 10x + 13\), and c) \((f \circ g)(-2) = -3\).

Step 1: Expand (g ○ f)(x)

The first composition we need to find is \((g \circ f)(x)\), which means that we need to plug the function \(f(x)\) into the function \(g(x)\). This is shown as \[ (g \circ f)(x) = g(f(x)), \]

Step 2: Simplify (g ○ f)(x)#tag_content#Now we simplify the expression:

\[ (g \circ f)(x) = -x^2 + 10x + 4 + 1 = -x^2 + 10x + 5. \] #tag_title

Step 3: Expand (f ○ g)(x)#tag_content#Now, we need to find the second composition, \((f \circ g)(x)\), which means we need to plug the function \(g(x)\) into the function \(f(x)\). This is shown as

\[ (f \circ g)(x) = f(g(x)). \] We expand this as follows: \[ (f \circ g)(x) = f(x+1) = -(x+1)^2 + 10(x+1) + 4. \] #tag_title

Step 4: Simplify (f ○ g)(x)#tag_content#Now we simplify the expression:

\[ (f \circ g)(x) = - (x^2 + 2x + 1) + 10x + 10 + 4 = -x^2 + 10x + 13. \] #tag_title

Step 5: Compute (f ○ g)(-2)#tag_content#Now we need to find the value of \((f \circ g)(-2)\), which is when \(x = -2\).

\[ (f \circ g)(-2) = -(-2)^2 + 10(-2) + 13 = - (-4) - 20 + 13 = -3. \] #tag_title#Answer#tag_content#The compositions are: a) \((g \circ f)(x) = -x^2 + 10x + 5\), b) \((f \circ g)(x) = -x^2 + 10x + 13\), and c) \((f \circ g)(-2) = -3\).

Key Concepts

Algebra Functions

Algebra functions are mathematical expressions that describe a relationship between two sets of numbers. They use operations like addition, subtraction, multiplication, and division. In the exercise you're working with, we have two functions: \(f(x)=-x^2+10x+4\) and \(g(x)=x+1\). These functions are written in terms of the variable \(x\), which is a placeholder for any number you can input.

Understanding algebra functions involves knowing how to work with these expressions and how to perform operations on them, such as combining them through addition or composition.

• **Polynomial Function:** The function \(f(x)\) is a quadratic polynomial because its highest exponent of \(x\) is 2.
• **Linear Function:** The function \(g(x)\) is a linear function because it’s just a straight line equation with \(x\) raised to the power of 1.

Both these functions can be interacted with to form something new, like composite functions, which we will discuss further.

Composite Functions

Composite functions involve combining two functions into a single function using composition. When you see notation like \((g \circ f)(x)\), it means you're creating a composite function where \(f(x)\) is inserted into \(g(x)\). Basically, you're taking the output from \(f(x)\) and using it as the input for \(g(x)\).

Composite functions are like nesting one function inside another. Here's how we work with them:

• **Functions In Series:** When asked to find \((g \circ f)(x)\), you start by calculating \(f(x)\) and then substitute this result into \(g(x)\).
• **Example:** For \((f \circ g)(x)\), you compute \(g(x)\) first, then plug that result into \(f(x)\).
• **Specific Value:** To find \((f \circ g)(-2)\), calculate \(g(-2)\) and use that output as the input to \(f\).

This method needs you to carefully perform each step, ensuring the correct order of operations, so pay close attention to details!

Function Notation

Function notation is a convenient way to express the equation of a function, showing clearly what variable the function is dependent upon. In our given functions, we use \(f(x)\) and \(g(x)\) to denote that both are functions of \(x\). Function notation helps us quickly identify what inputs are being used and what transformations they undergo to become outputs.

Here's how it simplifies working with functions:

• **Input and Output:** When you see \(f(x)\), think about it as an operation that you perform on \(x\). The result is the output.
• **Clear Communication:** Function notation clarifies what variable you’re using as an input, so you always know what output depends on.
• **Computational Aid:** It aids in composing functions because you can easily see which result is fed into the next function, like in \((g \circ f)(x)\) where \(f(x)\) is inserted into \(g(x)\).

Mastering function notation will give you a better grasp of how changes in inputs affect the outputs in mathematical models.