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Chapter 2: Problem 25

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following $$ f(2 t+1) $$

Short Answer

Expert verified
f(2t + 1) = -6t + 1

Step by step solution

01

- Identify the function

The given function is defined as: \[ f(x) = -3x + 4 \]
02

- Substitute the argument

Replace the variable \(x\) in the function \(f(x)\) with \(2t + 1\). This gives us: \[ f(2t + 1) = -3(2t + 1) + 4 \]
03

- Simplify the expression

Distribute the -3 through the parentheses: \[ -3(2t + 1) = -6t - 3 \] Then add 4 to the result: \[ -6t - 3 + 4 = -6t + 1 \]
04

- Final result

Combining the simplified terms, we have: \[ f(2t + 1) = -6t + 1 \]

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Key Concepts

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

Substitution
Substitution is a key concept in algebra that makes solving complicated equations easier. When working with functions, substitution involves replacing a variable with another expression. Imagine you have a function like \( f(x) = -3x + 4 \) and you want to evaluate this function at a new argument, such as \( 2t + 1 \). In this case, you would replace every instance of \( x \) in the function with \( 2t + 1 \). This process is called substitution and is very useful in transforming and solving functions.

Here鈥檚 what it looks like step-by-step:
  • Identify the original function. In our example, it鈥檚 \( f(x) = -3x + 4 \).
  • Replace \( x \) with \( 2t + 1 \) to get: \( f(2t + 1) = -3(2t + 1) + 4 \).
This technique helps simplify and solve equations involving functions through systematic replacements.
Simplifying Expressions
Simplifying expressions is another core concept in algebra. It helps in reducing complex equations into simpler, more manageable forms. Once substitution is done, the next step is to simplify the new expression. This often involves performing arithmetic operations and combining like terms. Consider our previous substituted expression: \( f(2t + 1) = -3(2t + 1) + 4 \).

To simplify this:
  • Distribute the multiplication across the parentheses: \( -3(2t + 1) \) becomes \( -6t - 3 \).
  • Then, combine the like terms by adding 4 to \( -3 \): \( -6t - 3 + 4 = -6t + 1 \).
The result, \( -6t + 1 \), is a simpler expression than what we started with. Simplifying expressions allows you to handle and solve algebraic problems more effectively.
Linear Functions
Linear functions are among the most basic and essential types of functions in algebra. They have the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. In the given exercise, \( f(x) = -3x + 4 \) is a linear function because it matches the form \( mx + b \).

Let鈥檚 break down the components:
  • \( m \), the coefficient of \( x \), represents the slope, which in this case is -3. It indicates that for every increase of 1 in \( x \), the function output decreases by 3.
  • \( b \), the constant term, is 4. It indicates where the line intersects the y-axis when \( x = 0 \).
Understanding the structure of linear functions helps in easier manipulation and solving, especially when combined with other concepts like substitution and simplifying expressions.

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Most popular questions from this chapter

The table represents a linear function. (a) What is \(f(2)\) ? (b) If \(f(x)=-1.3,\) what is the value of \(x ?\) (c) What is the slope of the line? (d) What is the \(y\) -intercept of the line? (e) Using the answers from parts (c) and (d), write an equation for \(f(x)\). $$ \begin{array}{c|c} x & y=f(x) \\ \hline 0 & 3.5 \\ \hline 1 & 2.3 \\ \hline 2 & 1.1 \\ \hline 3 & -0.1 \\ \hline 4 & -1.3 \end{array} $$

Graph each linear or constant function. Give the domain and range. $$ g(x)=4 x-1 $$

Three points that lie on the same straight line are said to be collinear. Consider the points \(A(3,1), B(6,2),\) and \(C(9,3) .\) Find the slope of segment \(B C\).

Graph the intersection of each pair of inequalities. $$ 6 x-4 y<10 \text { and } y>2 $$

Graph the union of each pair of inequalities. $$ x+y \leq 2 \quad \text { or } \quad y \geq 3 $$
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