Problem 39 - Algebra 2

Question

If \(f(x)=4 x, g(x)=2 x-1,\) and \(h(x)=x^{2}+1,\) find each value. $$ f[f(-3)] $$

Step-by-Step Solution

Answer

The value is -48.

Step 1: Find f(-3)

First, substitute -3 into the function \( f(x) = 4x \). \[ f(-3) = 4(-3) = -12 \] So, \( f(-3) = -12 \).

Step 2: Find f(f(-3))

Now, substitute the result from Step 1, which is \( f(-3) = -12 \), back into the function \( f(x) = 4x \).\[ f(-12) = 4(-12) = -48 \] Thus, \( f(f(-3)) = -48 \).

Key Concepts

Function Notation

Function notation is a way to represent functions in mathematical expressions. Functions are like machines where you input a number, and the machine processes it and gives you an output. The notation helps in identifying which function is being used and what variable or number is being input into it. For example, when you see something like \( f(x) = 4x \), it tells you that the function is named \( f \) and it operates on an input \( x \). The expression means that whatever number is plugged into \( x \), the function will multiply it by 4.
It's important to remember that different functions might have different rules for processing an input. This is indicated by the formula following the function notation. Some functions transform inputs through multiplication, as in \( f(x) = 4x \), while others might involve more complex operations like subtraction, as in \( g(x) = 2x - 1 \), or even exponentiation, like \( h(x) = x^2 + 1 \). Each function operates independently following its indicated operations.Understanding function notation is crucial because it allows us to easily communicate and manipulate mathematical functions, especially when dealing with composite functions, functions of functions, like \( f(f(-3)) \), where the output of one function becomes the input for another.

Evaluation of Functions

Evaluation of functions involves substituting specific values into the function and calculating the result. It is a critical skill because it helps us determine the output of a function for various inputs. In the exercise, you are asked to find \( f(f(-3)) \). To do this, you first evaluate \( f(-3) \).First, substitute \(-3\) into the function \( f(x) = 4x \). This involves multiplying \(-3\) by 4, resulting in \( f(-3) = -12 \). This is the first layer of evaluation, helping us break down the function's operation step by step.Next, evaluate \( f(-12) \) using the same function. Here, you substitute \(-12\) back into \( f(x) = 4x \), which requires multiplying \(-12\) by 4. This multiplication gives you \( f(-12) = -48 \), which is the final output for \( f(f(-3)) \).
Evaluation is like following a recipe – you perform each operation as indicated in the function definition, ensuring all steps are executed correctly to get the desired result.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the building blocks of functions. In the context of the given functions, expressions such as \( 4x \), \( 2x - 1 \), and \( x^2 + 1 \) describe how each function will process its input. Each expression follows specific rules that dictate the mathematical operations that need to be performed on the input value.The algebraic expression \( 4x \) used in the function \( f(x) = 4x \) is straightforward. It tells you to multiply the input by 4, a linear transformation. In contrast, the expression \( x^2 + 1 \) in \( h(x) = x^2 + 1 \) indicates a quadratic transformation, where the input is squared first and then increased by 1.Algebraic expressions allow functions to transform inputs systematically and predictably. When working with composite functions, understanding how each individual expression operates is crucial. This understanding enables you to correctly evaluate complex multi-step problems, similar to the original exercise where you had to perform multiple evaluations using the expression \( 4x \) twice.
An in-depth understanding of algebraic expressions ensures you can interpret and solve a wide range of functional problems effectively.