Problem 1 - Precalculus: Mathematical for Calculus

Question

If a function \(f\) is given by the formula \(y=f(x),\) then \(f(a)\) is the ________ of \(f\) at \(x=a\)

Step-by-Step Solution

Answer

Value

Step 1: Understanding the Function Notation

The notation \( y = f(x) \) is used to denote that \( y \) is a function of \( x \). This means that for each input \( x \), there is a corresponding output \( y \).

Step 2: Identifying the Question

We are asked to identify what \( f(a) \) represents in the context of the function \( f \) at \( x = a \).

Step 3: Evaluating the Function at a Specific Point

The expression \( f(a) \) indicates that we are substituting the value \( a \) into the function \( f \). Thus, \( f(a) \) gives us the output value of the function when the input is \( a \).

Step 4: Final Answer

Therefore, \( f(a) \) is the **value** of \( f \) at \( x=a \).

Key Concepts

Evaluating Functions

When we talk about evaluating functions, we're discussing the process of finding out what value a function will produce when a specific input is given. Functions are like machines that take an input and transform it into an output based on a set rule or formula. The notation \( f(x) \) is what we use to symbolize this function process. Here, \( x \) represents the input.

To evaluate a function, simply substitute the given input value into the function formula. If we're given a function defined as \( f(x) = 2x + 3 \), and we want to evaluate this function for \( x = 5 \), we replace \( x \) in the formula with 5. So, \( f(5) = 2(5) + 3 = 13 \). Therefore, the output for the input \( x = 5 \) is 13.

• This process allows us to find out exactly what the output is when we insert a particular input.
• The method is straightforward: replace \( x \) with the given number, then compute the result using basic arithmetic operations.

Function Value

In the realm of mathematics, the function value is the result you get from substituting a specific input into the function. In simpler words, it is the output of the function. This concept is important because it helps us determine what a function 'does' at any specified point in its domain.

Take for example a function represented as \( f(x) = x^2 - 4x + 5 \). If we want to find the function value when \( x = 3 \), we replace every \( x \) in the function formula with 3, giving us \( f(3) = 3^2 - 4(3) + 5 \). Calculating this results in \( f(3) = 9 - 12 + 5 = 2 \). Therefore, the function value at \( x = 3 \) is 2.

• The function value can change depending on the input.
• Understanding what each input maps to is crucial for analyzing the behavior of the function.

Input-Output Relationship

A fundamental principle in studying functions is the input-output relationship. This relationship is the heart of function notation and evaluation. Every function has an input, represented typically by \( x \), and an output, which is the result \( f(x) \).

Functions describe how each input relates to an output. Think of it like a recipe: you use certain ingredients (inputs), follow the recipe (function rule), and you get a dish (output). For instance, with \( f(x) = 3x - 2 \), if you input 4, you calculate \( f(4) = 3(4) - 2 = 10 \). Here, the input is 4 and the output is 10.

• Every input has a corresponding output, but different inputs can yield different outputs.
• The relationship defined by the function is what mathematicians use to model and solve real-world problems.