Problem 55 - College Algebra

Question

Find a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$ f(x)=4 x-3, g(x)-5 x^{2}-2 $$

Step-by-Step Solution

Answer

The solutions to each part are as follows: a. (f ∘ g)(x) = 20x^{2}-11, b. (g ∘ f)(x) = 80x^{2} -120x + 43, c. (f ∘ g)(2) = 69, d. (g ∘ f)(2) = 123

Step 1: Compute (f ∘ g)(x)

Substitute \(g(x)\) into \(f(x)\). So, \(f(g(x)) = 4(5x^{2}-2)-3 = 20x^{2}-8-3 = 20x^{2}-11\).

Step 2: Compute (g ∘ f)(x)

Substitute \(f(x)\) into \(g(x)\). So, \(g(f(x)) = 5(4x-3)^{2}-2 = 5(16x^{2}-24x+9)-2 = 80x^{2} -120x + 45 -2= 80x^{2} -120x + 43\).

Step 3: Compute (f ∘ g)(2)

Substitute \(2\) into \((f \circ g)(x)\). Thus, \(f(g(2)) = 20*2^{2}-11 = 20*4 -11 = 80-11 = 69\).

Step 4: Compute (g ∘ f)(2)

Substitute \(2\) into \((g \circ f)(x)\). Thus, \(g(f(2)) = 80*2^{2} -120*2+43 = 80*4-240+43 = 320-240+43 = 123\).

Key Concepts

Polynomial Functions

Polynomial functions are a fundamental class of mathematical expressions. They consist of variables and constants, combined using addition, subtraction, and multiplication, raised to whole number powers. For example, the function given in the exercise, \( g(x) = 5x^{2} - 2 \), is a polynomial function.

Key characteristics of polynomial functions include:

• Degree: The degree is determined by the highest power of \(x\) in the expression. For \( g(x) = 5x^{2} - 2 \), the degree is 2.
• Coefficients: These are the numerical factors multiplying the variable, like the 5 in \(5x^{2}\).
• Constant term: A term that does not contain any variables; in \(g(x)\), it is \(-2\).

Understanding these attributes helps when addressing operations such as function composition, as seen in the exercise.

Substitution Method

The substitution method is a straightforward technique for evaluating functions, especially when dealing with function composition. In this context, it involves replacing the variable in one function with another function. This is precisely what was done in the exercise.

For example, to find \((f \circ g)(x)\), you substitute \(g(x) = 5x^{2} - 2\) into \(f(x) = 4x - 3\). Thus, you calculate \(f(g(x)) = 4(5x^{2} - 2) - 3\). The result is simplified to \(20x^{2} - 11\).

Steps to apply substitution:

• Identify the 'inner' and 'outer' functions: Here, \(g(x)\) is the inner function, and \(f(x)\) is the outer function for the composition \((f \circ g)(x)\).
• Substitute and simplify: Replace \(x\) in the outer function by the expression for the inner function, then simplify the resulting expression.

This same method is used for \((g \circ f)(x)\) in the example, demonstrating versatility in problem-solving.

Evaluating Functions

Evaluating functions involves calculating the value of a function for a specific input. This is a skill essential not only in textbook problems but also in real-world applications.

In the given exercise, after determining \((f \circ g)(x)\) and \((g \circ f)(x)\), we are asked to evaluate these compositions at \(x = 2\).

Process for evaluation:

• Substitute the value: Replace \(x\) with 2 in the simplified compositions found earlier. For example, \(f(g(2)) = 20 \cdot 2^{2} - 11 = 69 \).
• Perform arithmetic operations: Ensure correct order of operations, combining like terms as needed.

This process confirms the value of the composed function at a particular point, essential in verifying solutions or in scenarios where specific numerical inputs are given.