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Chapter 7: Problem 19

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

Short Answer

Expert verified
f(x + h) = -3x - 3h + 4

Step by step solution

01

Identify the given function

The given function is: \( f(x) = -3x + 4 \).
02

Substitute \( x + h \) into \( f(x) \)

Replace every instance of \( x \) in \( f(x) \) with \( x + h \). This gives: \[ f(x + h) = -3(x + h) + 4 \]
03

Simplify the expression

Distribute the \( -3 \) inside the parenthesis: \[ f(x + h) = -3x -3h + 4 \].

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Key Concepts

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

Algebraic Expressions
An algebraic expression is a mathematical phrase that can involve numbers, variables, and operations like addition, subtraction, multiplication, and division. In algebraic expressions, variables represent unknown values and are often denoted by letters, such as x, y, or z. Expressions can be simple, like \(2x + 3\), or more complex, like \(x^2 - 4x + 7\).
Understanding how to manipulate these expressions is crucial when working with functions in algebra and calculus.
In our exercise, the main algebraic expression we are dealing with is \(f(x) = -3x + 4\). Algebraic expressions form the basis for creating and evaluating functions.
Function Evaluation
Function evaluation means finding the output of a function for specific input values. To evaluate a function, you substitute the input value for the variable in the function. Consider the function \(f(x) = -3x + 4\).
If you want to find \(f(2)\), you substitute 2 for every instance of x: \(f(2) = -3(2) + 4 = -6 + 4 = -2\).
This simple process is fundamental for tasks like graphing functions or solving for specific values.
For our exercise, evaluating the function involves substituting \(x + h\) for \(x\) to find \(f(x+h)\).
Substitution Method
The substitution method is a powerful tool used in algebra to simplify and solve equations. It involves replacing a variable with a given value or expression. In our example, we need to find \(f(x + h)\) when \(f(x) = -3x + 4\).
Using the substitution method, we replace \(x\) with \(x + h\):
\[f(x + h) = -3(x + h) + 4 \]

Next, we distribute and simplify the expression:
Distribute the \(-3\) through the parentheses:
\[ f(x + h) = -3x - 3h + 4 \]
Using substitution helps simplify complex problems and makes it easier to evaluate functions for specific inputs.

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