Problem 29 Let $f(x)=-3 x+4$ and \(g(x)=-... [FREE SOLUTION]

Chapter 2: Problem 29

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

Identify the function and variable substitution

The given function is $f(x) = -3x + 4$. To find $f(x+h)$, we need to substitute $x+h$ into the function in place of $x$.

Substitute $x+h$ into the function

Replace every occurrence of $x$ in $f(x)$ with $x+h$. Thus, we have: \[f(x+h) = -3(x+h) + 4\]

Simplify the expression

Distribute $-3$ through the term $x+h$: \[-3(x+h) = -3x - 3h\] Adding the constant term gives: \[f(x+h) = -3x - 3h + 4\]

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Key Concepts

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

Understanding Algebraic Functions

Algebraic functions are mathematical expressions that involve variables and constants. These functions use operations such as addition, subtraction, multiplication, and division. For instance, the given function in the exercise is an algebraic function:
$f(x) = -3x + 4$
It shows a relationship where each input value of $x$, when plugged into the function, gives an output. Here, the expression says 'multiply $x$ by -3 and then add 4'. Understanding how these functions behave with different inputs is essential for solving many algebra problems.

Concept of Variable Substitution

Variable substitution is a key concept in algebra. It involves replacing a variable in an expression with another variable or expression.
In the exercise, we need to find $f(x+h)$. This means we replace $x$ with the expression $x+h$ in our function:
1. Start with the original function: $f(x) = -3x + 4$
2. Substitute $x+h$ for $x$: $f(x+h) = -3(x+h) + 4$
This substitution helps us understand how the function changes when we adjust its input and is fundamental for solving complex algebra problems.

Simplification of Expressions

Simplification in algebra involves rewriting expressions in a more basic or clear form without changing their value. To simplify $-3(x+h) + 4$:

1. Distribute the -3 across the terms inside the parentheses: $-3(x+h) = -3x - 3h$
2. Combine like terms
Thus,$f(x+h) = -3x - 3h + 4$
By simplifying, we make the expression easier to understand and use further in solving algebraic problems. Simplification also helps in identifying patterns and properties in functions, which is crucial for deeper mathematical analysis.

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91影视

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

Short Answer

Step by step solution

Identify the function and variable substitution

Substitute \(x+h\) into the function

Simplify the expression

Key Concepts

Understanding Algebraic Functions

Concept of Variable Substitution

Simplification of Expressions

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