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Chapter 3: Problem 41

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. $$ 4 x-3 y=8 $$

Short Answer

Expert verified
The function is \(f(x) = \frac{4x - 8}{3}\). The value of \(f(3)\) is \(\frac{4}{3}\).

Step by step solution

01

Isolate the term with y

Start with the given equation: \[4x - 3y = 8\]Subtract 4x from both sides to isolate the term with y:\[-3y = 8 - 4x\]
02

Solve for y

Divide both sides of the equation by -3 to solve for y:\[y = \frac{8 - 4x}{-3}\]Simplify the expression:\[y = \frac{-8 + 4x}{3} = \frac{4x - 8}{3}\]
03

Write y as a function of x

Replace y with function notation \(f(x)\):\[f(x) = \frac{4x - 8}{3}\]
04

Substitute x = 3 into the function

To find \(f(3)\), substitute the value of x into the function:\[f(3) = \frac{4(3) - 8}{3}\]Simplify the expression:\[f(3) = \frac{12 - 8}{3} = \frac{4}{3}\]

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Key Concepts

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

Function Notation
Function notation is a way to write equations that makes it clear a variable (usually y) depends on another variable (usually x). Instead of writing an equation like y = 2x + 3, we write it as f(x) = 2x + 3. This tells us that y is a function of x.

Function notation has several advantages:
  • It is more concise and easier to read.
  • It explicitly shows the input-output relationship between variables.
  • It can handle more complex relationships and multiple variables.
For example, consider the equation y = (4x - 8)/3. Using function notation, we write it as f(x) = (4x - 8)/3, which clearly shows y is a function of x.
Solving for y
Solving for y in an equation means isolating the y variable on one side of the equation. This often involves algebraic manipulation, like addition, subtraction, multiplication, or division.

For the equation 4x - 3y = 8, we can follow these steps:
  • Subtract 4x from both sides: -3y = 8 - 4x
  • Divide everything by -3: y = (8 - 4x)/-3
  • Simplify the negative signs: y = (4x - 8)/3
Now, we've isolated y and simplified the equation. Following these steps helps express y in terms of x.
Substitution Method
The substitution method involves replacing a variable with a specific value or another expression. This technique is commonly used in algebra to find the value of one variable given another.

For example, to find f(3) in the equation f(x) = (4x - 8)/3, we substitute x with 3:
  • Replace x with 3: f(3) = (4(3) - 8)/3
  • Simplify the expression: f(3) = (12 - 8)/3
  • Finish the calculation: f(3) = 4/3
The substitution method provides a straightforward way to evaluate functions at specific points.
Simplification
Simplification involves reducing an equation or expression to its simplest form. This often makes solving equations or evaluating functions easier.

When simplifying, look for ways to combine like terms, factor common terms, or reduce fractions. For instance, with the equation (8 - 4x)/-3, we can simplify:
  • Distribute the negative sign: -8 + 4x
  • Write it as: (4x - 8)/3
Simplification helps make equations more manageable and reveals the underlying relationships between variables.

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Most popular questions from this chapter

Find an equation of the line that satisfies the given conditions. (a) Write the equation in slope-intercept form. (b) Write the equation in standard form. Through \((-1,3) ;\) parallel to \(-x+3 y=12\)

Find an equation of the line that satisfies the given conditions. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form. $$ \text { Through }(-5,4) ; \text { slope } \frac{1}{2} $$

A hill rises 30 ft for every horizontal \(100 \mathrm{ft}\). Which of the following express its slope (or grade)? (There are several correct choices.) A. 0.3 B. \(\frac{3}{10}\) C. \(3 \frac{1}{3}\) D. \(\frac{30}{100}\) E. \( \frac{10}{3}\) F. 30 G. \(30 \%\) H. \(-\frac{10}{3}\)

Suppose a factory can have no more than 200 workers on a shift, but must have at least 100 and must mamufacture at least 3000 units at minimum cost. The managers need to know how many workers should be on a shift in onder to produce the required units at minimal cost. Linear programming is a method for finding the optimal (best possible) solution that meets all the conditions for such problems. Let \(x\) represent the number of workers and y represent the mumber of units manufactured. Work Exercises \(47-52\) in order. Write three inequalities expressing the conditions given in the problem.

Segment PQ has the given coordinates for one endpoint P and for its midpoint M. Find the coordinates of the other endpoint \(Q .\) (Hint: Represent \(Q\) by \((x, y)\) and write two equations using the midpoint formula, one imolving \(x\) and the other involving \(y .\) Then solve for \(x\) and \(y .\) $$ P(2.5,1.75), M(3,2) $$
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