Problem 67

Question

Find $f(a), f(b+1),$ and $f(3 x)$ for the given $f(x)$ $$f(x)=2 x-5$$

Step-by-Step Solution

Verified

Answer

$f(a) = 2a - 5$; $f(b+1) = 2b - 3$; $f(3x) = 6x - 5$.

1Step 1: Find f(a)

To find $ f(a) $, substitute $ a $ into the function in place of $ x $. The function $ f(x) = 2x - 5 $ becomes $ f(a) = 2a - 5 $. This is your expression for $ f(a) $.

2Step 2: Find f(b+1)

To find $ f(b+1) $, substitute $ b+1 $ into the function in place of $ x $. The function $ f(x) = 2x - 5 $ becomes $ f(b+1) = 2(b + 1) - 5 $. Simplify this to $ f(b+1) = 2b + 2 - 5 = 2b - 3 $.

3Step 3: Find f(3x)

To find $ f(3x) $, substitute $ 3x $ into the function in place of $ x $. The function $ f(x) = 2x - 5 $ becomes $ f(3x) = 2(3x) - 5 $. Simplify this to $ f(3x) = 6x - 5 $.

Key Concepts

SubstitutionAlgebraic ExpressionSimplification

Substitution

The concept of substitution is crucial in function evaluation. It involves replacing a variable within a function with a specified value or expression. In the context of our exercise, we are asked to compute $ f(a) $, $ f(b+1) $, and $ f(3x) $ based on the given function $ f(x) = 2x - 5 $. Here's how substitution works:

For $ f(a) $, we replace $ x $ with $ a $, resulting in $ f(a) = 2a - 5 $.
For $ f(b+1) $, substitute $ b+1 $ into the function, resulting in $ f(b+1) = 2(b+1) - 5 $.
For $ f(3x) $, replace $ x $ with $ 3x $, which gives us $ f(3x) = 2(3x) - 5 $.

The substitution process is straightforward, yet it's important to be meticulous with algebraic operations to ensure accuracy.

Algebraic Expression

An algebraic expression, like $ 2x - 5 $, represents a mathematical phrase involving numbers, variables, and operations. It's crucial to understand each part of an expression to manipulate it correctly:

**Variables**, like $ x $, $ a $, or $ b $, act as placeholders. In our given function, $ x $ is the main variable.
**Coefficients** are the numerical factors multiplied by variables. In $ 2x $, 2 is the coefficient.
**Constants** are fixed values. In $ 2x - 5 $, minus 5 is the constant.

When evaluating $ f(x) $, understanding these components helps to correctly substitute and simplify the expression. For each function evaluation ($ f(a) $, $ f(b+1) $, $ f(3x) $), these terms rearrange around each substituted value.

Simplification

Simplification is the process of combining like terms and reducing an expression to its simplest form. After substitution, this step is essential to reach the final answer. Let’s see how this works in our examples:

For $ f(b+1) $, after substitution: $ f(b+1) = 2(b+1) - 5 $, distribute $ 2 $ to each term inside the parentheses: $ 2b + 2 $. Then, combine the constant terms $ 2 - 5 $ to get $ 2b - 3 $.
For $ f(3x) $, substitute and simplify: $ f(3x) = 2(3x) - 5 $, multiply $ 2 $ and $ 3x $ to get $ 6x $, and since there’s only one constant $ -5 $, end with $ 6x - 5 $.

Each simplification process highlights the importance of correctly applying arithmetic operations and recognizing patterns in algebra.

Problem 66

Problem 67

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