Problem 25 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The value of the expression (g ∘ h)(-2) is 16.

Step 1: Understand Function Composition

Function Composition is the process of using one function's output as the input of another function. So we first calculate the output of function h(x) and then use that as an input for function g.

Step 2: Evaluate h(-2)

Plugging -2 into the function h, we get: \( h(-2) = {(-2)}^2 + 4 = 4 + 4 = 8 \). So, when x = -2, h(x) = 8.

Step 3: Evaluate g(h(-2))

Now that we have h(-2), we can find g(h(-2)) by substituting h(-2) into function g. So,\( g(h(-2)) = g(8) = 2 * 8 = 16 \).

Key Concepts

Evaluation of Functions

When evaluating functions, it's crucial to understand the process of substituting inputs into functions to find their corresponding outputs. This can feel like a puzzle where each piece fits perfectly. To get the output of any function, begin by identifying the function's expression. For example, with \(h(x) = x^2 + 4\), you substitute the input value into the expression, just as if you were solving a regular algebraic equation.

Consider evaluating \(h(-2)\): Plug -2 into the function. Substitute \(-2\) for \(x\) in the equation, giving \((-2)^2 + 4\). Calculate it step-by-step: \((-2)^2 = 4\), then \(4 + 4 = 8\). Thus, \(h(-2) = 8\).

• This step of substitution is crucial to obtaining the right function outputs.
• Always double-check your arithmetic to ensure accuracy.

This method makes it clear what value the function displays when given specific inputs.

Algebraic Operations

Algebraic operations form the backbone of handling functions. Knowing how to perform these operations correctly ensures the accurate evaluation of functions. When dealing with functions, operations such as addition, multiplication, and squaring are commonplace.

Take the function \(g(x) = 2x\). Here, the operation performed is multiplication of the input by 2. If you're asked to find \(g(8)\), you multiply 8 by 2, a straightforward algebraic operation resulting in \(16\).

• The simple operation is made complex by understanding its application within functions.
• Always execute these operations in the prescribed order to maintain precision.

Mastering these operations allows you to decode more complex function compositions.

Function Inputs and Outputs

The concept of function inputs and outputs is pivotal in understanding function operations. A function processes a given input to produce a specific output through its defined operation.

For instance, the composition \(g \, \circ \, h\) means using the output of \(h\) as the input for \(g\). Let's see it in action: Start with evaluating \(h(-2) = 8\). Here, -2 is the input and 8 is the output. This output then serves as the input for function \(g\). Apply it, like so: \(g(8) = 2*8 = 16\). In this scenario:

• -2 is the initial input,
• 8 is the intermediate output serving as a new input,
• 16 is the final output of the function composition.

Grasping how inputs transform into outputs is key to solving function-related problems effectively. Each step links together, forming a continuous chain leading to your final answer.