Problem 60 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\((f \circ g)(-3) = 16\).

Step 1: Understand Composition of Functions

The composition of two functions, written as \((f \circ g)(x)\), means plugging the output of the function \(g\) into the function \(f\). In other words, first evaluate \(g(x)\) and use the result as the input for \(f(x)\).

Step 2: Evaluate g(x) at x = -3

Substitute \(x = -3\) into \(g(x) = x^2 + x\):\[g(-3) = (-3)^2 + (-3) = 9 - 3 = 6.\]Thus, \(g(-3) = 6\).

Step 3: Evaluate f at g(-3)

Now that we know \(g(-3) = 6\), substitute this value into \(f(x) = 3x - 2\):\[f(g(-3)) = f(6) = 3(6) - 2 = 18 - 2 = 16.\]

Step 4: Conclusion

Thus, \((f \circ g)(-3) = f(g(-3)) = 16\).

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the context of functions, these symbols often represent numbers. This allows mathematicians to express general rules and relationships economically.
In dealing with functions like \(f(x) = 3x - 2\) and \(g(x) = x^2 + x\), algebra helps us systematically transform and solve expressions by following the set rules. For example, when you substitute \(x = -3\) into the function, algebraic rules guide how to simplify and evaluate that expression.
Algebra supports understanding concepts like function composition and the evaluation of functions by providing a framework to manipulate symbols consistently to unveil solutions or even predict outcomes.

Evaluate Functions

Evaluating functions means finding the value of a function for a given input. When you have a function like \(f(x) = 3x - 2\), evaluating it at a particular value of \(x\) involves substituting that value into the function and simplifying.
For instance, if you are given \(f(x) = 3x - 2\) and you need \(f(6)\), you substitute 6 in place of \(x\):

• Multiply: \(3 \times 6 = 18\)
• Subtract: \(18 - 2 = 16\)

Thus, \(f(6) = 16\). Each function will have its particular rule, but the process of substitution and simplification remains the same.
Evaluating functions is a crucial step in understanding larger functions or concepts such as composition, where we evaluate one function and use its output for another.

Composition of Functions

The composition of functions involves merging two or more functions so that the output of one function becomes the input of another. With composition, the notation \((f \circ g)(x)\) represents applying the function \(g\) first, then applying \(f\) to the result of \(g(x)\).
Using the example from above, we first evaluate \(g(x) = x^2 + x\) with \(x = -3\):

• Calculate: \((-3)^2 + (-3) = 9 - 3 = 6\)

Then, we input the result, \(6\), into \(f(x) = 3x - 2\):

• Compute: \(3 \times 6 = 18\)
• Subtract: \(18 - 2 = 16\)

Therefore, \((f \circ g)(-3) = 16\).
Function composition effectively layers functions to create complex transformations, allowing us to explore a wider range of relationships and interactions between variables.