Problem 20 - College Algebra

Question

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

Step-by-Step Solution

Answer

a. \(f \circ g(x) = 15x - 18\), b. \(g \circ f(x) = 15x + 2\), c. \((f \circ g)(2) = 12\)

Step 1: Compute \(f \circ g(x)\)

This indicates a composition of functions which means you substitute function \(g(x)\) into function \(f(x)\). So, \(f \circ g(x) = f(g(x)) = f(3x - 4)\). Substitute \(g\) into \(f\) replacing each \(x\) in \(f(x)\) by \((3x - 4)\). Thus, we get \(5*(3x - 4) + 2\). Perform the multiplication and simplification to get the result. The result will be \(15x - 18\).

Step 2: Compute \(g \circ f(x)\)

Similar to Step 1, this indicates a composition of functions which means you substitute function \(f(x)\) into function \(g(x)\). So, \(g \circ f(x) = g(f(x)) = g(5x + 2)\). Substitute \(f\) into \(g\) replacing each \(x\) in \(g(x)\) by \((5x + 2)\). Thus, we get \(3*(5x + 2) - 4\). Perform the multiplication and simplification to get the result. The result will be \(15x + 2\).

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

To find this, substitute \(2\) into the result from Step 1, which is the function \(f \circ g(x)\), giving you \((f \circ g)(2) = 15*2 - 18\). Simplify this to find the final solution.

Key Concepts

Algebra

Algebra involves understanding and manipulating expressions using symbols and numbers according to fixed rules. It is the branch of mathematics where letters and symbols represent numbers or quantities in formulas and equations.
When dealing with function compositions, algebra becomes key as we perform operations using variables to derive expressions. It acts as the foundation enabling us to replace variable parts of functions and perform necessary calculations to solve the given problem.
In our exercise, we are composing functions, which involves algebraic manipulation to substitute one function into another, multiplying terms, and combining like terms.

Substitution

Substitution is the process of replacing a variable in an expression or equation with another expression or a value. This is the heart of computing compositions of functions like dealing with \((f \circ g)(x) \) and \((g \circ f)(x)\).
To evaluate a composite function, begin by taking the inner function—in this case, either \(g(x)\) or \(f(x)\)—and inserting it into the outer function's variable placeholder. For instance, in \((f \circ g)(x)\), substitute \(3x - 4\) from \(g(x)\) into each \(x\) of \(f(x)\), leading to the calculation of expression substitutions like \(f(3x-4) = 5(3x-4)+2\).
This step is vital because accurate substitution sets the stage for correct simplification and evaluation.

Simplification

Once substitution is complete, simplification helps reduce the expression to its simplest form. This process involves performing arithmetic operations and combining like terms to render the expression more manageable.
Using our substituted expression for \(f(3x-4) = 5(3x-4)+2\), you multiply out the terms to yield \(15x - 20 + 2 \).
Finally, combine the constants \(-20 + 2\) to arrive at \(15x - 18\).
Simplification is crucial because it not only tidy up equations but also facilitates easier evaluation in subsequent steps.

Evaluate Functions

Evaluation in function composition means calculating the output of the composed function by substituting a specific value into it. This step usually comes after simplification.
For instance, to evaluate \((f \circ g)(2)\), you start with the simplified form of \((f \circ g)(x)\)—in this case, \(15x - 18\). Substitute \(x\) with \(2\), resulting in \(15*2 - 18\).
Then, carrying out the arithmetic gives 30 - 18, which simplifies to \(12\).
This computation shows how the composed function transforms a specific input, giving a concrete numerical result.