Problem 66 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

\(f(a) = a - 5\), \(f(b + 1) = b - 4\), and \(f(3x) = 3x - 5\).

Step 1: Substitute for f(a)

We need to find the expression for \(f(a)\). Since \(f(x) = x - 5\), we substitute \(a\) for \(x\). Thus, \(f(a) = a - 5\).

Step 2: Substitute for f(b+1)

Next, we find \(f(b + 1)\). Again, substitute \(b + 1\) into \(f(x) = x - 5\). Hence, \(f(b + 1) = (b + 1) - 5 = b - 4\).

Step 3: Substitute for f(3x)

Lastly, we find \(f(3x)\). Substitute \(3x\) into \(f(x) = x - 5\). Therefore, \(f(3x) = 3x - 5\).

Key Concepts

Substitution Method

The substitution method is a handy technique used in solving equations and evaluating functions. It involves replacing variables in an expression with specific values or other expressions. For example, if we have a function \( f(x) = x - 5 \), and we want to find \( f(a) \), we substitute \( x \) with \( a \). This gives us \( f(a) = a - 5 \).

Using substitution, one can easily find specific outputs of a function for particular inputs. This is useful in both linear and non-linear functions. In algebra, it shows us how a change in the input affects the output.

This method requires keen attention to detail, as changing one part of the equation or expression affects others. When substituting, ensure all occurrences of the variable are replaced to maintain the integrity of the problem.

Linear Functions

Linear functions are among the simplest and most familiar types of functions in algebra, given in the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants.

In our solution, the function \( f(x) = x - 5 \) is a linear function with a slope \( m = 1 \) and a y-intercept \( b = -5 \).

• The graph of a linear function is always a straight line.
• The slope \( m \) indicates the steepness and direction of the line.
• The y-intercept \( b \) is the point where the line crosses the y-axis.

Linear functions are pervasive in many fields and fundamentally represent a constant rate of change. Understanding them is crucial for tackling more complex problems in mathematics and related disciplines.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition, subtraction, multiplication, division, etc.). They form the building blocks of algebra. In our exercise, expressions like \( f(a) = a - 5 \), \( f(b+1) = b - 4 \), and \( f(3x) = 3x - 5 \) are examples of algebraic expressions.

Key aspects of algebraic expressions include:

• Variables: Symbols that represent numbers or values. In our case, \( a \), \( b+1 \), and \( 3x \) are the variables used.
• Operations: Indicate the mathematical processes like addition (\(+\)) and subtraction (\(-\)).
• Constants: Fixed values, such as \(-5\) in our expressions.

Mastering algebraic expressions allows us to form and manipulate equations, a fundamental skill in solving mathematical problems. Understanding these components helps in simplifying, evaluating, and transforming expressions to find desired solutions.