Problem 54 - Precalculus : Building Concepts and Connections

Question

In Exercises \(49-66,\) let \(f(x)=x^{2}+x, g(x)=\sqrt{x},\) and \(h(x)=-3 x\) Evaluate each of the following. $$(h \circ g)(0)$$

Step-by-Step Solution

Answer

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

Step 1: Evaluate the inner function

Given \(g(x)=\sqrt{x}\). Substitute \(x=0\) into \(g(x)\) to get \(g(0)\). So \(g(0)=\sqrt{0}=0\).

Step 2: Substitution into the outer function

Now, substituting \(g(0)\) into \(h(x)\), you have \(h(g(0)) = h(0)\).

Step 3: Evaluate the outer function

Given that \(h(x) = -3x\), substitute \(x=0\) into \(h(x)\) to get \(h(0)\). So \(h(0) = -3*0 = 0\).

Key Concepts

Composite Functions

Understanding composite functions is essential for delving deeper into the world of mathematics. In essence, a composite function is created when one function is applied within another function. Here is the core idea: when you have two functions, say f and g, composing them will result in a new function denoted as \( (f \circ g)(x) \)—read as 'f composed with g of x'.

This process works by taking the output from the first function, g(x), and using that result as the input for the second function, f. Therefore, when evaluating \( (h \circ g)(0) \) as in the given exercise, we first find g(0) and then apply h to that result. It's like a math relay race, where the baton is passed from the output of g to the input of h.

In practical terms, to correctly perform function composition, you must:

• Evaluate the inner function using the given input.
• Use the result of the inner function as the input for the outer function.
• Evaluate the outer function with this new input to get the final result.

Inner and Outer Functions

When we talk about composite functions, terms 'inner' and 'outer' frequently pop up. These terms are crucial for grasping how function composition works. The inner function is the function that is applied first, and its output is fed into another function, which we call the outer function.

The inner function can be thought of as the starting point—it’s what you do first. For the exercise above, the inner function is g(x), and since we want to evaluate \( (h \circ g)(0) \) we first determine what g(0) is. It’s akin to preparing the ingredients before cooking a dish.

After the inner function’s work is done, the outer function comes into play. It takes the output from the inner function and performs its operation to produce the final dish, so to speak. In our exercise, after finding out that g(0) is zero, we then apply the outer function h to zero. The outer function essentially determines the ultimate outcome of the composite function.

Function Evaluation

Function evaluation might sound technical, but it’s simply the act of finding the output of a function given a specific input. Every time you plug a number into a function, you are evaluating it. This step is fundamental in both single functions and within composite functions.

Here’s how you evaluate a function:

• Replace the variable in the function expression with the given input value.
• Simplify the expression using proper mathematical rules.
• Observe the result – this is your function evaluated at the given input.

For example, in our exercise, we evaluated the function h(x) = -3x at the point x=0. By substituting and simplifying, we found that h(0) equals zero. This simple process is a powerful tool across all levels of mathematics and is used continuously in algebra, calculus, and beyond.