Problem 127 - Algebra and Trigonometry Real Mathematics, Real People

Question

Let \(f(x)=14 x-3\) and \(g(x)=8 x^{2} .\) Find the indicated value. \((f \circ g)(-1)\)

Step-by-Step Solution

Answer

The value of the function composition \(f \circ g(-1)\) is 109.

Step 1: Evaluate the Inner Function

In the composition \(f \circ g(-1)\), start by evaluating the inner function, that is, compute the value of \(g(-1)\). The function \(g(x)\) is defined by \(g(x)=8 x^{2}\). Therefore, \(g(-1)=8 * (-1)^{2}=8\).

Step 2: Evaluate the Outer Function

Now, substitute the result from the inner function \(g(-1)=8\) into the outer function \(f\). The function \(f(x)\) is defined by \(f(x)=14 x-3\). Therefore, \(f(g(-1))= f(8) = 14*8 -3 =109\). This represents the composition \(f \circ g(-1)\).

Step 3: Provide the final answer

The end result of the function composition \(f \circ g(-1)\) is 109.

Key Concepts

Inner Function Evaluation

When working with function composition, the first step is evaluating the inner function. In our problem, we start by computing the value of the function \(g(x)\) at \(x = -1\). Given that \(g(x) = 8x^2\), we substitute \(-1\) into this equation:

• \(g(-1) = 8 \times (-1)^2 = 8 \times 1 = 8\)

This result, 8, becomes vital for the next step. It's essential to handle the operations carefully, especially squaring negative values, to ensure accurate results. Evaluating the inner function correctly sets the foundation for the entire composition process.

Outer Function Evaluation

After finding the value of the inner function \(g(-1)\), we move on to the outer function \(f(x)\). We use the result from the inner evaluation, which is 8, to substitute into \(f(x)\). Given \(f(x) = 14x - 3\), we substitute \(x = 8\):

• \(f(8) = 14 \times 8 - 3\)
• \(= 112 - 3\)
• \(= 109\)

This step is crucial as it transforms the composed function into a simple problem of substitution and calculation. Understanding how the outer function relies on the inner function's output is central to mastering function compositions.

Substitution Method

The substitution method is an essential technique in evaluating function compositions. It involves replacing a variable in one function with another function's output. In our example, we evaluated \(g(-1)\) to find 8 and substituted it into \(f(x)\) instead of \(x\). This method simplifies complex problems by breaking them into manageable parts:

• Evaluate the innermost function first.
• Substitute the result into the next function.

This sequential approach makes complex compositions much more manageable. Practicing substitution enhances problem-solving skills and deepens understanding.

Algebraic Functions

Algebraic functions consist of operations involving variables and constants. In the given problem, both \(f(x) = 14x - 3\) and \(g(x) = 8x^2\) are algebraic because they encompass multiplication and addition/subtraction. Mastering the manipulation of these expressions is key to solving problems involving function compositions:

• Identify the type of algebraic operation used.
• Understand the impact of squaring, multiplying, and subtracting in context.

By thoroughly understanding algebraic functions, students can accurately evaluate and compose various functions, leading to greater mathematical insight.