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Chapter 2: Problem 24

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

Short Answer

Expert verified
f(x-2) = -3x + 10

Step by step solution

01

Recall the Function Definition

Recall that the function provided is given by: f(x) = -3x + 4.
02

Substitute Variable in Function

Replace the variable x in the function \(f(x)\) with \(x - 2\). This means we will find \(f(x - 2)\) by substituting \(x - 2\) into every instance of x in the function definition. Function becomes: f(x - 2) = -3(x - 2) + 4.
03

Distribute and Simplify

Distribute and simplify the expression obtained: f(x - 2) = -3(x - 2) + 4 = -3x + 6 + 4 = -3x + 10.

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Key Concepts

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

Substitution in Functions
Substitution in functions is a method to evaluate the function for a different input. Instead of using the variable typically found in the function, we replace it with another expression. This practice makes solving function-related problems straightforward and reliable.
To substitute into a function, follow these steps:
  • Step 1: Identify the given function. In our exercise, it's given by: \(f(x) = -3x + 4\).
  • Step 2: Identify the new expression to be substituted. For this problem, it's \(x - 2\).
  • Step 3: Replace every instance of the variable (\(x\)) in the function with the new expression. Thus, \(f(x - 2)\) becomes \(-3(x - 2) + 4\).
This method allows us to find the function's value for any shift in 'x' provided in the problem.
Linear Functions
Understanding linear functions is crucial because they form the foundation of many concepts in algebra. These functions are of the format \(f(x) = ax + b\), where:\
  • \(a\) is the slope of the line and determines its steepness.
  • \(b\) is the y-intercept, which is where the line crosses the y-axis.
In our example, \(f(x) = -3x + 4\), the function is linear:\
  • The slope \(a = -3\) implies that for every unit increase in \(x\), \(f(x)\) decreases by 3 units.
  • The y-intercept \(b = 4\) indicates the line crosses the y-axis at \(y = 4\).
Such insights help us understand the behavior of the function and predict its values at various points.
Algebraic Simplification
Algebraic simplification is the process of making an expression more straightforward and compact. This step was crucial in the exercise while substituting in the function.
After substituting \((x - 2)\), we needed to simplify the resulting expression \(-3(x - 2) + 4\):
  • Step 1: Distribute the -3 to both terms inside the parentheses: \(-3(x - 2) = -3x + 6\).
  • Step 2: Add the remaining constant: \(-3x + 6 + 4 = -3x + 10\).
By combining like terms and performing operations step-by-step, we arrive at the simplified function \(f(x - 2) = -3x + 10\). Simplification is essential to make interpretations and further calculations easier.

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