Problem 40 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The value is 2.

Step 1: Evaluate the Function f(x)

Given the function \(f(x) = 4x\), we need to find \(f\left(\frac{1}{4}\right)\). Substitute \(x\) with \(\frac{1}{4}\):\[f\left(\frac{1}{4}\right) = 4 \times \frac{1}{4} = 1\]So, \(f\left(\frac{1}{4}\right) = 1\).

Step 2: Evaluate the Function h(x) with f(x) Result

We now need to evaluate \(h(x)\) at the value found in Step 1, which is 1. Given \(h(x) = x^2 + 1\), substitute \(x\) with the value 1:\[h(1) = 1^2 + 1 = 1 + 1 = 2\]Thus, \(h(f(\frac{1}{4})) = 2\).

Key Concepts

Evaluating Functions

Evaluating functions is like cooking with precise ingredients; you need to follow the recipe correctly to get the desired result. When you evaluate a function, you substitute the input value with a given number. Imagine the function as a machine. You put a specific number into this machine, and it processes the number according to its formula to give you an output. For instance, with the function \(f(x) = 4x\), evaluating \(f\left(\frac{1}{4}\right)\) is like feeding \(\frac{1}{4}\) into the machine. This machine then multiplies \(\frac{1}{4}\) by 4, and out comes the number 1.

• Substitute the input: Replace \(x\) with the given input value.
• Apply the function rule: Perform the arithmetic operation according to the function’s formula.
• Get the result: The output is the function's value at the given input.

Next time you evaluate a function, just remember—input goes in, follow the formula, and output comes out.

Algebraic Functions

Algebraic functions are the backbone of many types of math problems. They consist of variables, numbers, and algebraic operations such as addition, subtraction, multiplication, and division. Let's take a look at some algebraic functions as seen in the problem:

• \(f(x) = 4x\) is a linear function, representing a straight line when graphed.
• \(g(x) = 2x - 1\) is another linear function with a different slope and y-intercept.
• \(h(x) = x^2 + 1\) is a quadratic function; its graph is a parabolic curve.

While solving for function composition, you often deal with multiple algebraic functions. As shown in the given solution, understanding these functions help to simplify and solve even more complex mathematical challenges like function compositions and transformations.

Function Notation

Function notation is a simple yet powerful way to describe functions and their operations clearly. Think of it as a label that tells you what the machine (the function) does to any number you put in. In function notation, \(f(x)\), \(g(x)\), and \(h(x)\) all represent different functions

• The letter (e.g., \(f, g, h\)) is just a name identifying the function.
• The symbol \(x\) represents the input variable or independent variable.
• The expression after the equal sign describes what the function does to \(x\).

When you see something like \(f(2)\), it prompts you to find the output by substituting 2 for every occurrence of \(x\) in the function’s formula. This notation helps make the evaluation process tidy and systematic, especially when dealing with multiple functions at once. What's great is that it scales gracefully from simple operations to more intricate compositions as you've seen in the solution.