Problem 30 - Intermediate Algebra

Question

Determine the indicated functional values. (Objective 2 ) If \(f(x)=-5 x+2\) and \(g(x)=-3 x^{2}+4\), find \((f \circ g)(-2)\) and \((g \circ f)(-1)\).

Step-by-Step Solution

Answer

(f ∘ g)(-2) = 42; (g ∘ f)(-1) = -143.

Step 1: Understand the Composition of Functions

To determine \((f \circ g)(x)\) and \((g \circ f)(x)\), we need to evaluate the functions at certain values in a specific sequence. \((f \circ g)(x)\) means we first find \(g(x)\) and then use that result in \(f(x)\). Similarly, \((g \circ f)(x)\) means we first find \(f(x)\) and use that result in \(g(x)\).

Step 2: Evaluate \(g(-2)\)

To find \((f \circ g)(-2)\), start by evaluating \(g(x)\) at \(x = -2\). Substitute \(-2\) into \(g(x) = -3x^2 + 4\):\[ g(-2) = -3(-2)^2 + 4 = -3(4) + 4 = -12 + 4 = -8 \]

Step 3: Evaluate \(f(g(-2))\)

With \(g(-2) = -8\), we now substitute \(-8\) into \(f(x) = -5x + 2\):\[ f(-8) = -5(-8) + 2 = 40 + 2 = 42 \]Thus, \((f \circ g)(-2) = 42\).

Step 4: Evaluate \(f(-1)\)

To find \((g \circ f)(-1)\), start by evaluating \(f(x)\) at \(x = -1\). Substitute \(-1\) into \(f(x) = -5x + 2\):\[ f(-1) = -5(-1) + 2 = 5 + 2 = 7 \]

Step 5: Evaluate \(g(f(-1))\)

With \(f(-1) = 7\), we substitute \(7\) into \(g(x) = -3x^2 + 4\):\[ g(7) = -3(7)^2 + 4 = -3(49) + 4 = -147 + 4 = -143 \]Thus, \((g \circ f)(-1) = -143\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in algebra. It involves finding the output of a function for a specific input value. Imagine a function as a machine where you insert an input, and the machine processes it to give you an output. For example, if we have a function \( f(x) = -5x + 2 \), and we want to evaluate it at \( x = -1 \), we replace \( x \) with \( -1 \) everywhere it appears in the expression:

• Substitute \( x = -1 \) into \( f(x) \):
• \( f(-1) = -5(-1) + 2 \)
• Simplify to find \( f(-1) = 5 + 2 = 7 \).

Similarly, evaluating \( g(x) = -3x^2 + 4 \) at \( x = -2 \) is done by substituting \( -2 \) for \( x \), giving us \( g(-2) = -3(-2)^2 + 4 \), which simplifies to \( -12 + 4 = -8 \). It is crucial to substitute the given values carefully and perform the arithmetic operations accurately.

Intermediate Algebra

Intermediate algebra is a stepping stone that builds upon basic algebra concepts, forming the foundation for more advanced topics in mathematics. It usually involves working with polynomials, quadratic equations, and function operations, like in this problem of composition of functions. Understanding how to substitute specific values into functions and evaluate them is an essential skill in intermediate algebra. For example, when dealing with function compositions, it helps to keep the order of operations in mind. First, evaluate the inner function and then the outer function. For \((f \circ g)(-2)\), this requires finding \(g(-2) = -8\) first, then plugging the result into \(f(x)\). This flow of operations, from finding outer compositions to simplifying expressions, is a key theme in intermediate algebra.

Function Operations

Function operations revolve around various ways functions can be combined or transformed. One of these operations is composition, where you apply one function to the result of another. The notation \((f \circ g)(x)\) translates to applying \(g(x)\) first, then using its result as input for \(f(x)\). To illustrate, for \((f \circ g)(-2)\), you first compute \(g(-2) = -8\), then use this output in \(f(x)\) to find \(f(-8) = 42\). Similarly, for \((g \circ f)(-1)\), compute \(f(-1) = 7\) first, and then substitute into \(g(x)\) to get \(g(7) = -143\).Key points to remember:

• Always follow the correct order: inner first, outer second.
• Ensure all substitutions are accurately calculated.
• Double-check arithmetic operations for errors.

Understanding these processes is crucial for mastering function operations and successfully solving composite function problems.