Problem 8 - Algebra 2

Question

If \(f(x)=3 x, g(x)=x+7,\) and \(h(x)=x^{2},\) find each value. $$ g[h(-2)] $$

Step-by-Step Solution

Answer

\( g[h(-2)] = 11 \).

Step 1: Evaluate h(-2)

To solve for \( h(-2) \), substitute \(-2\) into the function \( h(x) = x^2 \). This gives us \( h(-2) = (-2)^2 \). Simplify to find \((-2)^2 = 4\). Thus, \( h(-2) = 4 \).

Step 2: Evaluate g[h(-2)]

Now, substitute the result from Step 1 into the function \( g(x) = x + 7 \). We found that \( h(-2) = 4 \), so substitute \( 4 \) into \( g(x) \). Compute \( g(4) = 4 + 7 \) which simplifies to \( 11 \).

Key Concepts

Function Composition

Function composition is like making a sandwich. You start with a base, and then you layer on top. In mathematics, this concept involves creating a new function by combining two or more functions. Think of it like a sequence of operations applied to a value.

• The function that you start with is called the inner function.
• The function that you apply afterward is the outer function.

The original exercise demonstrates function composition with the functions \(h(x)\) and \(g(x)\). We first evaluated the inner function \(h(-2)\) giving us a result of \(4\). Then, we used this result as the input for the outer function \(g(x)\), ending up with a final value of \(11\).
So, in a nutshell, function composition involves nesting one function into another, like layers in a process. This is written as \(g(h(x))\), meaning you compute \(h(x)\) first, then \(g\) of that result.

Evaluating Functions

Evaluating functions is the process of finding the output of a function for a specific input value. It's like figuring out what happens to a number when you put it into a machine that follows a certain rule or formula.

• To evaluate a function, you substitute the given value into the function rule.
• Simplify the expression to find the result.

In our example, we started by evaluating \(h(-2) = (-2)^2\), which resulted in \(4\). This involved substituting \(-2\) into the formula \(x^2\) and calculating the result.
Then we took this result \(4\) and evaluated \(g(4)\), by substituting \(4\) into \(x + 7\), simplifying it to find the result \(11\). Evaluating functions is a straightforward technique of plugging in numbers to see what the function outputs.

Quadratic Functions

Quadratic functions are a type of polynomial function that can always be written in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. They create a graph that is a symmetric curve called a parabola.

• In the given example, \(h(x) = x^2\) is a simple quadratic function.
• This function only involves \(x^2\) and has no linear or constant term other than \(x^2\) itself.

Quadratic functions like \(h(x)\) often appear in problem-solving to transform inputs through squaring. When we evaluated \(h(-2)\), we used the squaring rule inherent in quadratic functions, resulting in \((-2)^2 = 4\).
Understanding quadratic functions is crucial as they feature widely in algebra and real-world scenarios, such as calculating areas, trajectories of objects, and more.