Problem 23 - Algebra 2

Question

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (h \circ g)(-5) $$

Step-by-Step Solution

Answer

The result of \( (h \circ g)(-5) \) is 104.

Step 1: Understanding Function Composition

The composition of two functions, denoted as \( (h \circ g)(x) \), means to plug the output of function g into function h.

Step 2: Applying g to x

First, calculate \(g(-5)\). Substituting \(-5\) for \(x\) in the function \(g\) gives \(g(-5)=2(-5)=-10\).

Step 3: Substituting into h

Substitute \(g(-5)=-10\) into function \(h\). This gives \(h(g(-5))=h(-10)=(-10)^{2}+4=104\).

Key Concepts

Algebraic Functions

Algebraic functions are a fundamental part of mathematics. These functions consist of variables and constants combined through algebraic operations like addition, subtraction, multiplication, and division. Algebraic functions can also include powers, as seen in the expression for the function \(h(x) = x^2 + 4\). Here, \(h(x)\) is defined by a squared term plus a constant. On the other hand, \(g(x) = 2x\) involves multiplication by a constant. Algebraic functions allow us to model real-life scenarios and solve various problems by manipulating equations. They are crucial in calculus, engineering, physics, and several other fields. Understanding how to work with algebraic functions is essential for tackling more complex mathematical problems."},{

Function Evaluation

Function evaluation is the process of calculating an output for a specific input value within a given function. To evaluate a function, you simply substitute the input value into the function's expression and perform the required calculations. For example, with the function \(g(x) = 2x\), to find \(g(-5)\), replace \(x\) with \(-5)\) to get \(g(-5) = 2(-5) = -10\).Similarly, evaluating the function \(h(x) = x^2 + 4\) at a particular value involves substituting that value in place of \(x\) and computing the result, as with \(h(-10)\). This results in \(h(-10) = (-10)^2 + 4 = 100 + 4 = 104\).Function evaluation is crucial for understanding how functions behave, predicting outcomes, and solving equations. It's a foundational skill for anyone working with mathematical functions.

Mathematical Operations

Mathematical operations are the basic actions we perform on numbers and algebraic expressions. These include addition, subtraction, multiplication, division, and exponentiation. In the context of algebraic functions, operations are used to transform inputs into outputs. For example, in the function \(g(x) = 2x\), multiplication is the operation transforming \(x\) into \(g(x)\).In our exercise, evaluating \((h \circ g)(-5)\) involves several operations:

• Multiplying \(-5\) by 2 to find \(g(-5) = -10\).
• Squaring \(-10\) to compute \((-10)^2 = 100\).
• Adding 4 to obtain the final result \(100 + 4 = 104\).

Understanding how to apply these operations is essential for solving complex problems, and mastering them will aid in your comprehension of all mathematical functions and concepts.