Problem 14 - Algebra A Combined Function

Question

If \(f(x)=x^{2}-6 x+2, g(x)=-2 x\), and \(h(x)=\sqrt{x}\), find each composition. $$ (h \circ g)(0) $$

Step-by-Step Solution

Answer

The composition \((h \circ g)(0) = 0\).

Step 1: Understand the Problem

We are asked to find the composition of the functions \(h\) and \(g\) evaluated at \(x=0\). This means we need to substitute the output of \(g(0)\) into \(h(x)\).

Step 2: Evaluate \(g(0)\)

Substitute \(0\) into the function \(g(x)=-2x\):\[g(0) = -2(0) = 0\]

Step 3: Evaluate \(h(g(0))\) or \(h(0)\)

Using the result from Step 2, substitute \(0\) into the function \(h(x)=\sqrt{x}\):\[h(0) = \sqrt{0} = 0\]

Step 4: Conclude with Final Result

The value of the composition \((h \circ g)(0)\) is equal to 0.

Key Concepts

Evaluation of Functions

When we talk about evaluating functions, we're essentially looking at how we can find the value of a function for a particular input. A function can be thought of as a machine that takes an input, processes it in a certain way, and then gives an output.
In mathematical terms, if we have a function like \( g(x) = -2x \), and we want to evaluate it at \( x = 0 \), we substitute 0 for \( x \), resulting in \( g(0) = -2(0) = 0 \).
Evaluating functions is a key skill in algebra and calculus. It allows us to understand how functions behave under different conditions and find specific outputs for given inputs. Being proficient in this process helps us tackle more complex mathematical problems involving function compositions and transformations.

Substitution

The concept of substitution is often used to simplify expressions or solve equations by replacing a variable with a specific value or another expression. In function composition, substitution is necessary to evaluate the compounded effect of multiple functions.
When dealing with function compositions like \( (h \circ g)(0) \), we first find the result of one function and then substitute this result into another function. In the example we have, after evaluating \( g(0) = 0 \), we use this outcome as the input for the next function, \( h(x) \).
Substitution helps break down complex calculations into simpler steps, making it easier to manage multiple operations and understand how multiple functions interact. It is a valuable tool for students to handle intricate equations with confidence.

Square Root Function

A square root function is one which involves finding the number that, when multiplied by itself, gives the original number. This function is defined as \( h(x) = \sqrt{x} \).
Evaluating a square root function involves taking a number and determining its principal square root—a non-negative value whose square is equal to the original number. For example, to evaluate \( h(0) \) where \( h(x) = \sqrt{x} \), we find that \( h(0) = \sqrt{0} = 0 \).
Understanding square root functions is integral to many areas of mathematics, including geometry and algebra. Knowing how to correctly evaluate these functions ensures that you can solve a range of problems, from simple equations to more complex polynomial expressions. It also highlights the importance of function properties in maintaining correct and consistent results across different scenarios.