Problem 21 - College Algebra

Question

In Exercises \(21-32,\) evaluate each function at the given values of the independent variable and simplify. $$f(x)=4 x+5$$ a. \(f(6)\) b. \(f(x+1) \quad\) c. \(f(-x)\)

Step-by-Step Solution

Answer

The simplified answers for these three functions are \(f(6) = 29\), \(f(x+1) = 4x + 9\), and \(f(-x) = -4x + 5\).

Step 1: Substitution of \(x=6\)

In the first function we substitute \(x\) with 6, so it becomes \(f(6) = 4*6 + 5\). Before simplifying the term, ensure the correct substitution has been made.

Step 2: Simplifying \(f(6)\)

After substituting we simplify the function, \(f(6) = 4*6 + 5 = 24 + 5 =29\).

Step 3: Substitution of \(x+1\)

In the second function instance, we substitute \(x\) with \(x+1\), so it becomes \(f(x+1) = 4*(x+1) + 5\). Ensure the use of parentheses while substituting to preserve the correct expression.

Step 4: Simplifying \(f(x+1)\)

After substituting, the expression should be simplified: \(f(x+1) = 4x + 4 + 5 = 4x + 9 \).

Step 5: Substitution of \(-x\)

In the third function instance, we replace \(x\) with \(-x\), so it becomes \(f(-x) = 4*(-x) + 5\). Again, it's important to use parentheses to get the right expression.

Step 6: Simplifying \(f(-x)\)

After substituting, we get and simplify the expression: \(f(-x) = -4x + 5\).

Key Concepts

Substitution in Functions

Understanding the concept of substituting values into functions is essential for evaluating algebraic expressions. Substitution involves replacing the variable, which is often referred to as the independent variable, with a specific value or another expression. When we talk about a function like f(x) = 4x + 5, the function describes a relationship where every input x gives us a unique output determined by the formula.

To evaluate a function for a specific value, such as f(6), we simply substitute 6 in place of x. This looks like: f(6) = 4 * 6 + 5. The result is a numerical answer that represents the value of the function when x is 6. Substitution is also used to understand how a function behaves for a more general input like x+1 or -x. In these cases, you replace x with the entire expression (x+1) or (-x), ensuring to use parentheses to maintain the correct order of operations. Substitution lays the groundwork for exploring functions and how they vary with different inputs.

Simplifying Expressions

Once we have substituted the values or expressions into the function, the next step is simplifying the expression to find a more concise form. Simplifying might include combining like terms, performing arithmetic operations, or applying algebraic properties such as the distributive property.

Take for instance f(x+1). After the substitution step, we get f(x+1) = 4*(x+1) + 5. To simplify, we distribute the 4 to both x and 1, resulting in 4x + 4. Then we add 5 to get 4x + 9. Similarly, for f(-x), simplicity arrives at f(-x) = -4x + 5 after substituting -x into the function. Simplifying expressions is a key skill in algebra that allows us to discover the most straightforward form of an algebraic expression.

Algebraic Operations

Algebraic operations include the basic arithmetic operations—addition, subtraction, multiplication, and division—applied to algebraic expressions. Beyond these, operations like exponentiation, and in contexts with functions, operation of composition and inversion also fall under the umbrella of algebraic operations.

In the context of function evaluation, as seen with f(x)=4x+5, algebraic operations are used at every step of the process. After substitution, we multiply and add within the function to simplify the expression. It's crucial to adhere to the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction). Always apply multiplication before addition, as in f(6) = 4*6 + 5. Mastery of algebraic operations is vital for simplifying expressions optimally and solving for variables within algebraic equations, a foundational aspect of algebra.