Problem 27 - Intermediate Algebra

Question

Determine the indicated functional values. (Objective 2 ) If \(f(x)=9 x-2\) and \(g(x)=-4 x+6\), find \((f \circ g)(-2)\) and \((g \circ f)(4)\).

Step-by-Step Solution

Answer

(f ∘ g)(-2) = 124, (g ∘ f)(4) = -130.

Step 1: Understand Composition of Functions

The composition of two functions is denoted by \((f \circ g)(x)\), which means \(f(g(x))\). Similarly, \((g \circ f)(x)\) is \(g(f(x))\). We need to plug the output of the inner function into the outer function.

Step 2: Find g(x) for x = -2

Calculate \(g(-2)\) by substituting \(-2\) into \(g(x) = -4x + 6\).\[g(-2) = -4(-2) + 6 = 8 + 6 = 14\]

Step 3: Find f(g(-2))

Now, substitute \(g(-2) = 14\) into \(f(x) = 9x - 2\) to find \(f(14)\).\[f(14) = 9(14) - 2 = 126 - 2 = 124\]

Step 4: Find f(x) for x = 4

Calculate \(f(4)\) by substituting \(4\) into \(f(x) = 9x - 2\).\[f(4) = 9(4) - 2 = 36 - 2 = 34\]

Step 5: Find g(f(4))

Substitute \(f(4) = 34\) into \(g(x) = -4x + 6\) to find \(g(34)\).\[g(34) = -4(34) + 6 = -136 + 6 = -130\]

Key Concepts

Understanding Functional Values

Functional values refer to the output values that a function gives when specific inputs are substituted into it. Think of a function as a machine that takes an input, performs a calculation, and spits out an output. For example, if you have the function \( f(x) = 9x - 2 \), and you want to find \( f(4) \), you simply substitute \( 4 \) for \( x \) and calculate. This process gives you the functional value for that specific \( x \).

• To find functional values, determine what the function does: Multiply the input by 9 and subtract 2.
• Use the chosen input, perform the operations, and find the answer.

This basic principle lies at the heart of working with algebraic functions, which are mathematical expressions used to relate two quantities. Calculating the functional value helps you understand what a function predicts or models for certain inputs.

Grasping Composite Functions

Composite functions can be thought of like a series of operations: one function feeds into another. The notation \((f \circ g)(x)\) means "first apply \( g \), then apply \( f \) to the result." It's a two-step process where nested operations are performed. If you start with \( g(x) = -4x + 6 \) and then apply \( f(x) = 9x - 2 \), each output from \( g \) becomes an input to \( f \).To find \((f \circ g)(-2)\):

• Calculate \( g(-2) \) to find \( 14 \).
• Substitute \( 14 \) into \( f \), which results in \( f(14) = 124 \).

Similarly, for \((g \circ f)(4)\):

• First calculate \( f(4) \) to find \( 34 \).
• Then substitute \( 34 \) into \( g \), resulting in \( g(34) = -130 \).

Understanding composite functions helps in tasks where you need to perform multiple operations in succession, revealing dependencies between different processes.

An Insight into Algebraic Functions

Algebraic functions are expressions constructed using algebraic operations such as addition, subtraction, multiplication, division, and exponentiation with respect to constant powers. Both \( f(x) = 9x - 2 \) and \( g(x) = -4x + 6 \) are algebraic functions defined in simple linear terms. They transform each input \( x \) in a straightforward manner using linear adjustments—constructing a line when graphed.Key insights:

• Linear functions like these involve a constant rate of change, represented by the coefficients \( 9 \) and \(-4\).
• The constants \(-2\) and \(6\) adjust the graph's position on the y-axis.
• Algebraic functions model many real-world situations where proportional relationships hold.

By comprehending algebraic functions, you build the foundation needed for more complex mathematical analysis and problem-solving in calculus and beyond.