Problem 26 - Algebra 2

Question

Let \(g(x)=2 x\) and \(h(x)=x^{2}+4 .\) Evaluate each expression. $$ (g \circ h)(0) $$

Step-by-Step Solution

Answer

The value of \( (g \circ h)(0) \) is 8.

Step 1: Evaluate h(x) at x = 0

The function \(h(x) = x^{2} + 4\). Substitute \(x = 0\) into \(h(x)\), we get \(h(0) = (0)^{2} + 4 = 4\).

Step 2: Evaluate g(h(0))

Now substitute the value obtained from \(h(0)\) in Step 1, which is 4, into \(g(x) = 2x\). This gives us: \(g(h(0)) = g(4) = 2(4) = 8\).

Key Concepts

Evaluating Functions

Understanding how to evaluate functions is a key skill you need to master in algebra. Evaluating a function simply means to find the function's value at a certain input.
To do this, you'll substitute the given input into the function and simplify.
For example, if you have a function like \( h(x) = x^2 + 4 \) and you need to evaluate \( h(0) \), you substitute 0 wherever you see \( x \) in the function.

• Substitution: Replace \( x \) with 0: \( h(0) = 0^2 + 4 \).
• Calculation: Simplify this to get \( 0 + 4 = 4 \).

By performing this simple step of substitution and calculation, you can evaluate any function at any given point.
This forms the basis of many operations you'll perform on functions.

Algebraic Functions

Algebraic functions involve expressions made up of constants and variables using algebraic operations such as addition, subtraction, multiplication, division, and exponentiation.
In our example, we have two such algebraic functions: one is \( g(x) = 2x \) and the other \( h(x) = x^2 + 4 \).
These functions take an input, like \( x = 0 \), perform an algebraic operation on it, and produce an output.

• For \( g(x) = 2x \), it multiplies the input by 2.
• For \( h(x) = x^2 + 4 \), it squares the input and then adds 4.

Algebraic functions are foundational in mathematics because they describe real-world relationships and changes in quantities.
They are building blocks for more complex equations and analyses in mathematics.

Operation on Functions

Operations on functions allow you to create new functions by combining existing ones in various ways.
One of the important operations is function composition, denoted as \((g \circ h)(x)\).
This means you first apply the function \(h\) and then apply function \(g\) to the result.

• Start with the inner function: Evaluate \( h(x) \) using your given input, such as 0 in our exercise.
• Use the result: Insert the output of \( h(x) \) as input for \( g(x) \).

For the expression \((g \circ h)(0)\):

• First, find \( h(0) = 4 \).
• Use \( 4 \) in \( g(x) \) to find \( g(4) = 8 \).

Function operations like this allow us to build complex relationships from simpler ones, adding depth to the analysis and problem-solving in mathematics.