Problem 53 - Algebra 2

Question

FIND THE ERROR Danette and Marquan are trying to find \([g \circ f](3)\) for \(f(x)=x^{2}+4 x+5\) and \(g(x)=x-7 .\) Who is correct? Explain your reasoning.

Step-by-Step Solution

Answer

The correct value of \\[g \circ f](3)\] is 19.

Step 1: Understand Composition of Functions

The composition of two functions \(f\) and \(g\), denoted as \([g \circ f](x)\), is defined as \(g(f(x))\). This means we first apply \(f\) to the input and use the result as input for \(g\).

Step 2: Evaluate f at x = 3

Given \(f(x) = x^2 + 4x + 5\), we substitute \(x = 3\) to compute \(f(3)\).\[f(3) = 3^2 + 4(3) + 5 = 9 + 12 + 5 = 26\]

Step 3: Evaluate g at f(3)

Now we need to compute \(g(f(3))\), which is \(g(26)\), using the function \(g(x) = x - 7\).\[g(26) = 26 - 7 = 19\]

Step 4: Conclusion

The value of \([g \circ f](3)\) is 19. Check Danette's and Marquan's answers to see who is correct.

Key Concepts

Evaluating Functions

When diving into the world of functions, one of the first tasks you'll encounter is evaluating them. This simply means finding the output of a function for a given input. Suppose you have a function, say \( f(x) = x^2 + 4x + 5 \). To evaluate this function at \( x = 3 \), we substitute \( 3 \) into the function in place of \( x \).

This gives us:

• First, compute \( 3^2 = 9 \).
• Next, calculate \( 4 \times 3 = 12 \).
• Finally, sum up all components: \( 9 + 12 + 5 = 26 \).

So, \( f(3) = 26 \). The key here is to follow the given operations step by step sequentially, ensuring you handle arithmetic as per standard order of operations (parentheses, exponents, multiplication and division, addition and subtraction).

By clearly understanding each component of the function, you ensure accuracy in your computations.

Composition Notation

Function composition is a concept where the output of one function becomes the input of another. This is often denoted using composition notation, \([g \circ f](x)\), and represents \( g(f(x)) \). It essentially means you first apply the function \( f \) to \( x \), and then use the result as the input for \( g \).

Here's how it works with our example functions:

• Start with \( f(x) = x^2 + 4x + 5 \).
• For \( x = 3 \), we have already found \( f(3) = 26 \).
• Next, take \( g(x) = x - 7 \) and substitute \( f(3) \) for \( x \), giving \( g(26) = 26 - 7 = 19 \).

This step-by-step process is crucial in correctly evaluating composition, ensuring that you first find each interior function's value before moving to the outer function. Always remember, with \([g \circ f](x)\), \( f(x) \) is your starting point, and you build from there.

Error Checking

In mathematical operations, especially when handling complex notions like function composition, error checking is a vital step. It helps verify that you have correctly followed steps and arithmetic to reach the right solution. Here's how you can incorporate error checking into function evaluations:

- **Double-Check Substitutions:** Ensure that each value you substitute into the function is correctly placed. For example, when calculating \( f(3) \), verify each component: \( 3^2 + 4 \times 3 + 5 \).- **Review Arithmetic:** Pay close attention to the addition, multiplication, and any exponentiation, confirming that each part was calculated correctly.- **Verify Each Step:** When composing functions, like \([g \circ f](3)\), check that the output of \( f(3) \) was accurately used as the input for \( g(x) \).
For our problem, ensure that \( f(3) = 26 \) and \( g(26) = 19 \) are both correct. Error checking builds confidence in your answer and reduces mistakes, especially under pressure or when dealing with complex problems.