Problem 58 - Intermediate Algebra

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+x .\) Find each of the following. $$ (g \circ f)(4) $$

Step-by-Step Solution

Answer

(g \circ f)(4) = 110.

Step 1: Understand the composition of functions

The notation \((g \circ f)(4)\) means we need to find \(g(f(4))\). This involves first evaluating the inner function \(f(x)\) at \(x = 4\) and then using that result to evaluate \(g(x)\).

Step 2: Evaluate \(f(4)\)

Substitute \(x = 4\) into the expression for \(f(x)\). We have: \(f(x) = 3x - 2\). Therefore, \(f(4) = 3(4) - 2 = 12 - 2 = 10\).

Step 3: Evaluate \(g(10)\)

Now that we know \(f(4) = 10\), substitute \(x = 10\) into the expression for \(g(x)\), which is \(g(x) = x^2 + x\). Therefore, \(g(10) = 10^2 + 10 = 100 + 10 = 110\).

Step 4: Write the final answer

Since \(g(f(4)) = g(10) = 110\), we conclude that \((g \circ f)(4) = 110\).

Key Concepts

Evaluating Functions

Evaluating functions means finding the output of a function for a given input. To evaluate a function, you substitute the input value into the function’s equation. Let's consider a function like \(f(x) = 3x - 2\). If you need to evaluate \(f\) at \(x = 4\), replace \(x\) with \(4\) in the equation. Here’s how it works:

• Start with the equation: \(f(x) = 3x - 2\).
• Substitute \(x = 4\): \(f(4) = 3(4) - 2\).
• Calculate the result: \(12 - 2 = 10\).

So, \(f(4) = 10\). This process is simple and helps determine how a function changes for specific inputs. Evaluating is a crucial first step in more complex operations like composition.

Polynomial Functions

Polynomial functions are expressions involving sums of powers of variables multiplied by coefficients. They appear often in mathematics and have many different forms. For instance, \(g(x) = x^2 + x\) is a polynomial function. Let’s break down this example:

• The highest power of \(x\) is \(2\), which makes it a quadratic function.
• The terms are \(x^2\) and \(x\) with coefficients of \(1\) for both.
• It is expressed in standard form as \(ax^2 + bx + c\), where \(a = 1\), \(b = 1\), and \(c = 0\).

Polynomial functions can be evaluated just like simpler functions. Substituting a specific \(x\) value lets you find the output. For example, if you substitute \(x = 10\) in \(g(x) = x^2 + x\), you'd get \(g(10) = 10^2 + 10 = 110\). This evaluates the function at that point, yielding a specific result.

Composition of Functions

The composition of functions is a method for combining two functions. Here, one function’s output becomes the input for another. This is commonly represented as \((g \circ f)(x)\), meaning apply \(f(x)\) first, then \(g\). It’s like using a tool within another tool. For example, with \(f(x) = 3x - 2\) and \(g(x) = x^2 + x\):

• First, evaluate \(f(x)\) at a point. Suppose \(x = 4\): we have \(f(4) = 10\).
• Then use this output as the input for \(g(x)\). So here \(g(10) = 10^2 + 10 = 110\).
• The final result for \((g \circ f)(4)\) is \(g(f(4)) = 110\).

This method lets you blend functions to create new outputs. It’s useful for complex models and analysis, simplifying multiple operations into one seamless expression.