Problem 11 - Introductory Algebra for College Students

Question

Evaluate each function at the given values. \(f(x)=7 x\) a. \(f(10)\) b. \(f(-4)\) c. \(f(0)\)

Step-by-Step Solution

Answer

The function \(f(x)=7x\) evaluates to \(f(10) = 70\), \(f(-4) = -28\), and \(f(0) = 0\) at x equals 10, -4, and 0 respectively.

Step 1 - Substitute the first value

To find \(f(10)\), substitute \(x = 10\) into the function to get \(f(10) = 7 (10)\).

Step 2 - Result for the first value

Simplify the equation to find that \(f(10) = 70\).

Step 3 - Substitute the second value

Evaluation of \(f(-4)\) requires substituting \(x = -4\) in \(f(x)=7x\) to get \(f(-4) = 7 (-4)\).

Step 4 - Result for the second value

Simplify the equation to get \(f(-4) = -28\).

Step 5 - Substitute the third value

To find \(f(0)\), substitute \(x = 0\) into the function to get \(f(0) = 7 (0)\).

Step 6 - Result for the third value

Simplify to get \(f(0) = 0\).

Key Concepts

Function Evaluation

Function evaluation is an essential skill in mathematics that involves finding the output of a function for given input values. It is a straightforward process which helps students understand the relationship between the input and output of a function. In essence, you're simply swapping the function's variable with a specific number to find out what happens when that variable takes on that value.

For example, if you have the function \(f(x) = 7x\), and someone asks you to evaluate it at \(x = 10\), you're expected to substitute \(10\) in place of \(x\) in the function. This substitution allows us to find out what the function equals when \(x\) is \(10\).

• Identify the variable in the function.
• Replace ("evaluate") the variable with the given number.
• Perform the arithmetic to find the result.

Function evaluation is often the first step in solving function-based problems. It serves as a bridge between abstract function notation and numerical calculations.

Substitution Method in Functions

The substitution method in functions is a simple yet powerful arithmetic operation used to evaluate functions. It involves substituting a specific number, given in the exercise or problem, into the function in place of its variable. This is done in a bid to calculate the value of the function at that specific point.

Here's a breakdown of how substitution works using the provided exercise:

• Start with the function: \(f(x) = 7x\).
• For \(f(10)\), replace \(x\) with \(10\) to get \(f(10) = 7(10)\). Simplifying, we find that \(f(10) = 70\).
• For \(f(-4)\), substitute \(x = -4\) into the function to get \(f(-4) = 7(-4)\), which simplifies to \(f(-4) = -28\).
• Lastly, for \(f(0)\), we substitute \(x = 0\), resulting in \(f(0) = 7(0) = 0\).

The substitution method is reliable for dealing with both numerical values and more complex expressions, making it a key concept in algebra and calculus. It provides a basis for understanding how changes in inputs affect the output of functions.

Algebraic Functions

Algebraic functions are a type of function defined by polynomial expressions. These can include variables raised to powers, constants, and products of numbers and variables. Understanding algebraic functions is vital, as they form the foundation of many mathematical concepts.

In our example, the function \(f(x) = 7x\) is a simple linear algebraic function, where you have:

• A constant multiplier (7) and a variable \(x\).
• The expression represents the class of linear functions, characterized by a single variable term and a constant multiplier.

Such functions depict straight-line graphs and offer insight into proportional relationships between variables. Learning to work with algebraic functions involves:

• Understanding how to manipulate these expressions to evaluate outputs.
• Applying substitution to replace variables with given numbers for calculations.

By mastering simple operations in algebraic functions, students are better prepared to tackle more complex equations and problems in mathematics.