Problem 112 - College Algebra Essentials

Question

Does \(f(x)\) mean \(f\) times \(x\) when referring to a function \(f ?\) If not, what does \(f(x)\) mean? Provide an example with your explanation.

Step-by-Step Solution

Answer

No, \(f(x)\) does not mean \(f\) times \(x\), it represents the function \(f\) applied to the input \(x\). \(f(x)\) is the output of the function \(f\) when \(x\) is the input. For example, in the function \(f(x) = 3x + 2\), \(f(1) = 5\), which is the result of applying the function to the input \(1\).

Step 1: Explain Functions Definition

A function is a rule that assigns to each element from one set (the domain) exactly one element in another set (the codomain). Therefore, \(f(x)\) represents a function \(f\) that takes \(x\) as an input and provides corresponding output based on the defined rule of function \(f\).

Step 2: Clarify Misunderstanding About Multiplication

Unlike multiplication, where \(c \cdot d\) means 'c times d', \(f(x)\) does not mean 'f times x'. Here, \(f\) is not a number or a variable, instead it's a function or a rule. \(x\) is an input to that rule and \(f(x)\) is the corresponding output value.

Step 3: Give An Example

For example, consider the function \(f(x) = 3x + 2\). Here \(x\) is the input and multiplying it by 3 and adding 2 gives the output \(f(x)\). For \(x = 1\), the output of function f is \(f(1) = 3 \cdot 1 + 2 = 5\). Notice that \(f(1)\) is not 'f times 1' but it's the output of function \(f\) when \(1\) is the input.