Problem 9 - Introductory Algebra for College Students

Question

Evaluate each function at the given values. \(f(x)=x+5\) a. \(f(7)\) b. \(f(-6)\) c. \(f(0)\)

Step-by-Step Solution

Answer

The values for \(f(7)\), \(f(-6)\) and \(f(0)\) are 12, -1 and 5 respectively.

Step 1: Evaluate \(f(7)\)

To evaluate \(f(7)\), substitute \(x=7\) in the function \(f(x)=x+5\). You will get \(f(7) = 7 + 5 = 12.\)

Step 2: Evaluate \(f(-6)\)

To evaluate \(f(-6)\), substitute \(x=-6\) in the function \(f(x)=x+5\). You will get \(f(-6) = -6 + 5 = -1.\)

Step 3: Evaluate \(f(0)\)

To evaluate \(f(0)\), substitute \(x=0\) in the function \(f(x)=x+5\). You will get \(f(0) = 0 + 5 = 5.\)

Key Concepts

Function Evaluation

When studying algebra, one of the fundamental skills that students must develop is understanding how to evaluate functions. Function evaluation involves finding the output of a function for a particular input. The input, usually represented by a variable like 'x', takes on different values that are substituted into the function equation to calculate the corresponding output.

For example, with the function \( f(x) = x + 5 \), evaluating it at various values requires substituting those values in place of 'x'. Therefore, when asked to find \( f(7) \), the value 7 is substituted into the function, yielding \( f(7) = 7 + 5 = 12 \). This process is not only crucial in algebra but also across various fields of mathematics where functions are used to model relationships between quantities.

Substitution Method

The substitution method is widely used in algebra to evaluate functions. It's a technique where we replace variables within an expression with the values provided and perform the arithmetic operations according to the rules of algebra.

When evaluating \( f(-6) \) with our given function \( f(x) = x + 5 \), we apply the substitution method by replacing 'x' with -6, leading to \( f(-6) = -6 + 5 = -1 \). This method is not only essential for working with functions but is also a key skill in solving systems of equations, simplifying expressions, and integrating algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of letters (variables) and numbers using arithmetic operations like addition, subtraction, multiplication, and division, along with exponents and roots. These expressions can be as simple as \( x + 5 \) or as complex as one might encounter in higher levels of algebra.

Understanding how to manipulate these expressions is necessary for evaluating functions. For instance, in the case of \( f(0) \) with the function \( f(x) = x + 5 \), the expression simplifies to \( f(0) = 0 + 5 = 5 \). Students must become adept at working with algebraic expressions to thrive in their study of algebra and related mathematical fields.