Problem 59 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\((g \circ f)(-3) = 110\).

Step 1: Identify Functions

We are given the functions \(f(x) = 3x - 2\) and \(g(x) = x^2 + x\). We need to compute \((g \circ f)(-3)\), which means evaluating \(g(f(-3))\).

Step 2: Calculate \(f(-3)\)

Substitute \(-3\) into the function \(f\). So, \(f(-3) = 3(-3) - 2\). Calculate this to find \((-9) - 2 = -11\).

Step 3: Substitute \(f(-3)\) into \(g(x)\)

Now that we know \(f(-3) = -11\), substitute \(-11\) into the function \(g(x)\). So, \(g(-11) = (-11)^2 + (-11)\).

Step 4: Evaluate \(g(-11)\)

Calculate \((-11)^2 = 121\) and add \(-11\). So, \(g(-11) = 121 - 11 = 110\).

Step 5: Conclusion

Thus, \((g \circ f)(-3)\) evaluates to 110.

Key Concepts

Evaluating Functions

Evaluating functions means finding the output value when a specific input is plugged into the function. Functions are like machines that take an input, perform some operations on it, and then produce an output.
The function is often represented as \(f(x)\), where \(x\) is the input.
When evaluating a function like \(f(x) = 3x - 2\), you plug in a specific number for \(x\). For example, to find \(f(-3)\), you replace \(x\) with \(-3\) and calculate as follows:

• Multiply \(3\) by \(-3\) to get \(-9\).
• Subtract \(2\) from \(-9\) to arrive at \(-11\).

The result, \(-11\), is the output of the function for this specific input.

Substitution in Functions

Substitution in functions is a method where we replace a variable with a specific value or another expression.
This is crucial in composition of functions where one function is substituted into another.
Let's look at the example of substituting one function into another using \(g(f(x))\). First, you need to evaluate \(f(-3)\). We already found that \(f(-3) = -11\).
Next, you take this output from the first function and use it as the input for the second function \(g(x)\).

• Substitute \(-11\) for \(x\) in \(g(x) = x^2 + x\).
• Calculate \((-11)^2 + (-11)\) to find the output.

This process of substituting and evaluating leads to finding the result of a compounded function, like \((g \circ f)(-3)\).

Square of a Number

The square of a number is simply the number multiplied by itself.
It's a critical operation in many mathematical functions, including the one we've worked with in this exercise: \(g(x) = x^2 + x\). When we calculate \((-11)^2\), we multiply \(-11\) by \(-11\).
It's important to remember that the square of a negative number is positive; therefore, \((-11) \times (-11)\) equals \(121\).
Understanding how to correctly square a number ensures accurate calculations in function evaluations, especially when combining operations such as in the expression \(x^2 + x\).