Problem 36 - Elementary and Intermediate Algebra

Question

Review function notation and domains of functions $$ \text { If } f(x)=4 x-7, \text { find } f(a+h) $$

Step-by-Step Solution

Answer

f(a + h) = 4a + 4h - 7

Step 1: Understand the Function

The given function is \( f(x) = 4x - 7 \). This is a linear function where the output is determined by multiplying the input by 4 and then subtracting 7.

Step 2: Substitute the Input

To find \( f(a + h) \), substitute \( a + h \) wherever \( x \) appears in the function definition. This means replacing \( x \) with \( a + h \) in \( f(x) = 4x - 7 \).

Step 3: Apply the Substitution

By substituting \( a + h \) into the function, we get: \( f(a + h) = 4(a + h) - 7 \).

Step 4: Simplify the Expression

Distribute the 4 through the parentheses: \( 4(a + h) = 4a + 4h \). Thus, \( f(a + h) = 4a + 4h - 7 \).

Key Concepts

Linear Functions

Linear functions are among the simplest and most important types of functions in mathematics. They have the general form: \( f(x) = mx + b \), where: • \( m \) represents the slope of the line. This tells you how steep the line is. • \( b \) is the y-intercept, the point at which the line crosses the y-axis. Linear functions create a straight line when graphed. Their simplicity makes them a fundamental concept in algebra and calculus. In the original problem, we have a linear function \( f(x) = 4x - 7 \). Here, the slope (m) is 4, and the y-intercept (b) is -7. This tells us that for every 1 unit increase in \( x \), the function's value increases by 4 units.

Domain of Functions

The domain of a function is the set of all possible input values (usually \( x \) values) that the function can accept. For linear functions like \( f(x) = 4x - 7 \), the domain is all real numbers. This means you can plug any real number into the function, and it will produce a valid output. In our problem, when we substitute \( a + h \) into the function, we didn't change the function's domain—it remains all real numbers. It's important to check the function's definition to ensure you're using inputs within its domain.

Substitution in Functions

Substitution in functions involves replacing the variable \( x \) within a function with another expression. This allows us to evaluate the function with different inputs. In our exercise, to find \( f(a+h) \) for the function \( f(x) = 4x - 7 \), we substitute \( a + h \) for \( x \). Here's the process: • Start with the original function: \( f(x) = 4x - 7 \). • Replace \( x \) with \( a + h \): \( f(a + h) = 4(a + h) - 7 \). • Distribute the 4: \( f(a + h) = 4a + 4h - 7 \). Now you've successfully used substitution to find a new expression for the function. This method is essential for working with more complex functions and finding their values at specific points.