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Chapter 3: Problem 77

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x-2)\) $$\begin{array}{rrrrrr}x & -4 & -2 & 0 & 2 & 4 \\\f(x) & 5 & 2 & -3 & -5 & -9\end{array}$$

Short Answer

Expert verified
g(0)=2, g(2)=-3, g(4)=-5; g(-4) and g(-2) cannot be determined.

Step by step solution

01

Understand the Relationship

We are given the function equation \( g(x) = f(x-2) \). This means that to find \( g(x) \), we must use the values of \( f(x) \) by adjusting \( x \) by adding 2, i.e., substituting \( x-2 \) into \( f \).
02

Calculate g(x) for Each x Value

To compute \( g(x) \) at specific points, we simply replace the points in the definition of \( g(x) = f(x-2) \). For example, to find \( g(-4) \), calculate \( f(-4-2) = f(-6) \), but since \( f(-6) \) is not provided, we can't determine \( g(-4) \) from the given data.
03

Practical Application

Evaluate \( g(x) \) using available \( f(x) \) values:- \( g(0) = f(0-2) = f(-2) = 2 \)- \( g(2) = f(2-2) = f(0) = -3 \)- \( g(4) = f(4-2) = f(2) = -5 \).Unavailable \( f \) values imply that \( g(-4) \) and \( g(-2) \) cannot be determined from the given data.
04

Construct the Numerical Representation of g(x)

Based on available calculations, the function \( g(x) \) can be represented numerically as follows:- \( g(0) = 2 \)- \( g(2) = -3 \)- \( g(4) = -5 \)This is based on the corresponding \( f(x) \) values provided for the respective shifted \( x \, values \).

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Key Concepts

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

Numerical Representation
Numerical representation refers to expressing a function in terms of specific numerical inputs and outputs. To understand this better, think of a function table as a map. The "x" values are the locations you want to visit, and "f(x)" values are the scenes at each location. In our exercise, we are given a table of values for function \(f\) at different locations:
  • \((-4, 5)\)
  • \((-2, 2)\)
  • \((0, -3)\)
  • \((2, -5)\)
  • \((4, -9)\)
These pairs show how each x-value corresponds to an output. This representation is crucial for understanding the behavior of function \(f\). It's a simple way to see particular points on a graph, giving us a quick snapshot of the function's behavior without needing a full graph.
Function Evaluation
Function evaluation is the process of determining the output of a function given an input. It helps in understanding how two functions relate to each other. In our exercise, the function \(g(x)\) is defined in terms of \(f(x)\) as \(g(x) = f(x-2)\). Each x-value in \(g\) is calculated by "shifting" the input of \(f\) by 2 to the right.
For instance, to evaluate \(g(0)\):
  • Calculate \(f(0-2)\), meaning you find \(f(-2)\).
  • From the table, \(f(-2) = 2\), so \(g(0) = 2\).
By repeating this process, you can find outputs for different inputs in \(g(x)\). If the required input for \(f\) is not listed in the table, like \(f(-6)\) for \(g(-4)\), the function \(g\) cannot be evaluated for those specific points based on the given data.
Shifted Functions
Shifted functions are transformations where the entire graph of a function moves in a specific direction. In the context of our exercise, \(g(x) = f(x-2)\) illustrates a "horizontal shift." This means you take the graph of \(f(x)\) and shift it 2 units to the right to get \(g(x)\).
Here's how horizontal shifts work:
  • A positive constant inside the function \(f(x-c)\) shifts the graph to the right by "c" units.
  • So, \(g(x) = f(x-2)\) means a rightward shift of 2 units.
  • If it were \(f(x+2)\), the graph would shift leftward by 2 units.
This shifting process doesn't alter the shape of the function graph; it only changes its position on the x-axis. Consequently, recognizing shifts is essential in predicting function behaviors without re-plotting complex graphs.

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Most popular questions from this chapter

Solve. Write answers in standard form. $$ 3 x^{2}-4 x=x^{2}-3 $$

Use transformations to sketch a graph of \(f\). \(f(x)=\sqrt{-x}\)

Braking distance for cars on level pavement can be approximated by \(D(x)=\frac{x^{2}}{30 k^{*}}\) The input \(x\) is the car's velocity in miles per hour and the output \(D(x)\) is the braking distance in feet. The positive constant \(k\) is a measure of the traction of the tires. Small values of \(k\) indicate a slippery road or worn tires. (Source: L. Haefner, Introduction to Transportation Systems.) (a) Let \(k=0.3 .\) Evaluate \(D(60)\) and interpret the result. (b) If \(k=0.25,\) find the velocity \(x\) that corresponds to a braking distance of 300 feet.

Use transformations to sketch a graph of \(f\). \(f(x)=\frac{1}{2}|x|\)

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x+1)-2\) $$\begin{array}{rrrrrrr}x & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline f(x) & 2 & 4 & 3 & 7 & 8 & 10\end{array}$$
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