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Chapter 2: Problem 50

Let \(f(x)=2 x-3\) and \(g(x)=-x+3 .\) Find each function value. $$(g \circ f)(3)$$

Short Answer

Expert verified
(g \circ f)(3) = 0

Step by step solution

01

Understand the composition of functions

Recall that \( (g \circ f)(x) = g(f(x)) \). This means you first apply the function \( f \) to \( x \) and then apply the function \( g \) to the result of \( f(x) \).
02

Find f(3)

Start by finding the value of \( f(3) \). Using the function \( f(x) = 2x - 3 \), substitute \( x \) with 3: \[ f(3) = 2(3) - 3 = 6 - 3 = 3 \]
03

Apply g to f(3)

Next, apply the function \( g(x) = -x+3 \) to the value found in Step 2, which is \( f(3) = 3 \). Substituting \( x \), we get: \[ g(3) = -(3) + 3 = -3 + 3 = 0 \]

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Key Concepts

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

Function Evaluation
Function evaluation is a fundamental concept in mathematics. It involves finding the output of a function for a given input. Think of it as feeding a number into a machine, which performs some operations and then spits out a result.

To evaluate a function, you simply substitute the given value into the function's equation and perform the arithmetic.

For example, with the function provided, \( f(x) = 2x - 3 \), if you want to find \( f(3) \), you substitute 3 in place of \( x \):
  • Step 1: Substitute \( x = 3 \)
  • \( f(3) = 2(3) - 3 \)
  • Step 2: Perform the operations: \( 6 - 3 = 3 \)
Now you know that \( f(3) = 3 \). This straightforward process is at the heart of many mathematical operations you'll encounter in precalculus and beyond.
Function Composition
The concept of function composition entails applying one function to the results of another function. In mathematical notation, composition is often represented as \( (g \, \circ \, f)(x) \), which reads as 'g of f of x'.

In our example, we are given two functions \( f(x) = 2x - 3 \) and \( g(x) = -x + 3 \). To find \( (g \, \circ \, f)(3) \), you first evaluate the inside function (\( f \)) with the given input and then use its result as the input to the outer function (\( g \)).

This can be broken down into:
  • Step 1: Evaluate \( f(3) = 3 \) (as previously discussed)
  • Step 2: Use this result as the input for function \( g \): \( g(3) = -(3) + 3 = 0 \)
  • Therefore, \( (g \, \circ \, f)(3) = 0 \)
Composition of functions can help solve complex problems by breaking them into smaller, more manageable steps.
Precalculus Operations
Precalculus involves bridging the gap between basic arithmetic/algebra and more advanced topics in calculus. Key operations include functions, their compositions, transformations, and evaluations.

When dealing with functions, you must understand how to evaluate them properly and how to compose them for more complex operations. This includes:
  • Understanding how to substitute values into functions.
  • Performing arithmetic operations accurately.
  • Understanding how changing inputs affects outputs, which is crucial for graphing functions and discussing their behavior.
These operations are foundational skills needed to progress to higher mathematical concepts where functions are not only evaluated but also integrated, differentiated, and optimized.

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Most popular questions from this chapter

The tables give some selected ordered pairs for functions \(f\) and \(g\). $$\begin{array}{|c|c|c|c|}\hline x & 3 & 4 & 6 \\\\\hline f(x) & 1 & 3 & 9 \\\\\hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|}\hline x & 2 & 7 & 1 & 9 \\\\\hline g(x) & 3 & 6 & 9 & 12 \\\\\hline\end{array}$$ Find each of the following. $$(g \circ g)(1)$$

For the pair of functions defined, find \((f+g)(x),(f-g)(x),(f g)(x),\) and \(\left(\frac{f}{g}\right)(x)\) Give the domain of each. $$f(x)=6-3 x, g(x)=-4 x+1$$

In Exercises 27 and \(28,\) use the table to evaluate each expression in parts ( \(a\) )- \((d),\) if possible. (a) \((f+g)(2)\) (b) \((f-g)(4)\) (c) \((f g)(-2)\) (d) \(\left(\frac{f}{g}\right)(0)\) $$\begin{array}{|c|c|c|}\hline x & f(x) & g(x) \\\\\hline-2 & -4 & 2 \\\\\hline 0 & 8 & -1 \\\\\hline 2 & 5 & 4 \\\\\hline 4 & 0 & 0 \\\\\hline\end{array}$$

Give the slope and y-intercept of each line, and graph it. $$y-\frac{3}{2} x-1=0$$

Given functions \(f\) and \(g,\) find ( \(a\) ) \((f \circ g)(x)\) and its domain, and ( \(b\) ) \((g \circ f)(x)\) and its domain. See Examples 6 and 7 . $$f(x)=x^{3}, \quad g(x)=x^{2}+3 x-1$$
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