Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-8,\) for each function: a. Write an expression for \(f(x) .\) b. Find \(f(5)\) \(x \stackrel{\mathrm{f}}{\rightarrow} x^{2}\)

Step-by-Step Solution

Answer

The expression is \( f(x) = x^2 \) and \( f(5) = 25 \).

Step 1: Define the Function

The given mapping is \( x \stackrel{\mathrm{f}}{\rightarrow} x^{2} \). This means that the function, \( f(x) \), maps any input \( x \) to \( x^2 \). Therefore, the expression for the function is \( f(x) = x^2 \).

Step 2: Substitute and Evaluate

We need to find \( f(5) \). Substitute \( x = 5 \) into the function \( f(x) = x^2 \). This gives us \( f(5) = 5^2 \).

Step 3: Calculate the Power

Calculate \( 5^2 \) which equals \( 25 \). Therefore, \( f(5) = 25 \).

Key Concepts

Understanding Function Notation

Function notation is a concise and efficient way to express the relationship between inputs and outputs in mathematical functions. It allows us to easily understand how a particular function operates. Here, we are focusing on the function notation \( f(x) \), where \( f \) is the name of the function and \( x \) is the variable. It essentially means that \( f \) takes \( x \) as an input and gives an output based on the rule defined by the function.

• In \( f(x) = x^2 \), \( x^2 \) is the rule that the function follows.
• When you see \( f(x) \), think of it as a machine: you put \( x \) in, and \( x^2 \) comes out.

Function notation helps in quickly recognizing what operation needs to be performed on the input to obtain the output. It also provides a foundation for more complex operations you might encounter in functions.

Substitution in Functions

Substitution in functions is the process of replacing the variable with a specific value to evaluate the function. This is a critical step in understanding how the function behaves for particular inputs. Once you have the function defined, such as \( f(x) = x^2 \), you can find the result of the function for any specific value by substituting the value in place of \( x \).

• Example: To find \( f(5) \), you replace \( x \) with \( 5 \).
• This becomes \( f(5) = 5^2 \).

Substitution lets you investigate how the function produces output from various inputs. Practicing this will help in mastering function operations, including more advanced functions later.

Evaluating Quadratic Functions

Evaluating quadratic functions involves substituting a given value into the quadratic equation and performing the required calculations. A quadratic function is typically expressed as \( ax^2 + bx + c \). However, in simpler terms, like our function \( f(x) = x^2 \), the equation is just \( x^2 \). Here's how you evaluate it for any specific input:

• First, substitute the input value into the function.
• Next, compute the power, which means multiplying the number by itself in the case of \( x^2 \).

For \( f(x) = x^2 \), and given \( x = 5 \), you evaluate \( f(5) \) as follows: compute \( 5^2 \), resulting in \( 25 \). This demonstrates the simplicity of calculations once the substitution is complete. Evaluating quadratic functions can reveal how inputs affect outputs, which is a critical part of understanding linear transformations in mathematics.