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Chapter 3: Problem 127

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

Short Answer

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

Step by step solution

01

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, \(g(-1)=8 * (-1)^{2}=8\).
02

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 \(f \circ g(-1)\).
03

Provide the final answer

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$g(x)=\frac{x^{2}-8 x+12}{x^{2}+4}$$

Divide using long division. $$\left(x^{2}-10 x+15\right) \div(x-3)$$

Divide using long division. $$\left(x^{2}+5 x+6\right) \div(x-4)$$

Simplify the expression. $$\left(\frac{x}{8}\right)^{-3}$$

A real zero of the numerator of a rational function \(f\) is \(x=c .\) Must \(x=c\) also be a zero of \(f ?\) Explain.
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