Problem 35 - 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

4a - 7 + h

Step 1: Understand the Function Notation

The function is given as \( f(x) = 4x - 7 \). This means that for any input \(x\), the function will return \(4x - 7\).

Step 2: Substitute the Input \(a\) into the Function

Replace \(x\) with \(a\) in the function notation \(f(x)\). Therefore, \( f(a) = 4a - 7 \).

Step 3: Add \(h\) to the Function Output

To find \( f(a) + h \,\) add \(h\) to the result from Step 2. Thus, \( f(a) + h = (4a - 7) + h \).

Step 4: Combine Like Terms

Add \(-7\) and \(h\) together. The final expression is \( f(a) + h = 4a - 7 + h \).

Key Concepts

Understanding Domains of Functions

In mathematics, the domain of a function is the set of all possible inputs for the function. For the function given in the exercise, \( f(x) = 4x - 7 \), the input can be any real number.

Determining the domain of a function requires checking if there are any restrictions on the inputs, such as:

• Values that could make the denominator zero (for rational functions).
• Values that could give negative results under a square root (for some root functions).

For the function \( f(x) = 4x - 7 \), there are no such restrictions, so the domain is all real numbers, often written as \( (-fty, +fty) \). Ensuring you understand the domain helps avoid errors when evaluating the function and combining terms.

Mastering Substitution

Substitution is a fundamental concept in algebra, where you replace a variable with a given value or expression. In this exercise, we replaced \( x \) with \( a \) in the function \( f(x) = 4x - 7 \).

When performing substitution:

• Identify the variable to substitute.
• Ensure you replace every instance of the variable in the expression.
• Simplify after substitution as much as possible.

In the exercise, substituting \( a \) into \( f(x) \) transformed the function into \( f(a) \), giving us \( f(a) = 4a - 7 \). This step is crucial before proceeding to the final expression involving combining like terms and operations.

Combining Like Terms

Combining like terms is the process of simplifying algebraic expressions by merging terms that have the same variables raised to the same power. In the exercise given, after substituting \( a \) into the function, we obtained \( f(a) = 4a - 7 \). The next step was adding \( h \) to it.

To combine like terms:

• Identify terms that have identical variable parts.
• Add or subtract the coefficients of these terms.
• Leave constants unchanged unless combined with other constants.

In our example, we end up with \( f(a) + h = 4a - 7 + h \). Here, \( -7 \) and \( h \) are constants and cannot be further combined as they don't share variables. The final simplified form is thus \( f(a) + h = 4a - 7 + h \).

This step helps create a straightforward expression for further calculations or evaluations.