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Chapter 5: Problem 41

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5\). \((f \circ g)(0)\)

Short Answer

Expert verified
(f \circ g)(0) = 13

Step by step solution

01

- Find g(0)

First, evaluate the function g at x = 0. The function g is given by: \(g(x) = 2x + 3\) Substitute 0 for x: \(g(0) = 2(0) + 3 = 3\)
02

- Use g(0) as the input for f

Next, we need to find \((f \circ g)(0)\), which means we have to evaluate the function f at the value obtained from g(0). From Step 1, \(g(0) = 3\). Now we use this as the input for f: \(f(x) = x^2 + 4\) Substitute 3 for x: \(f(3) = 3^2 + 4 = 9 + 4 = 13\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

composite functions
In mathematics, composite functions are like a chain of commands. You first perform one function then use its result as the input for another function.
This linked process is what we call 'function composition.' When we write \((f \circ g)(x)\), we mean 'first apply g to x, then apply f to the result of g(x).' This way, composite functions help us combine simpler functions into more complex operations.
Using our example, let's walk through the composite function \(f \circ g)(0)\):
  • First, calculate the inner function g at x = 0, so we find g(0).
  • Then, take this result and use it as the input for the outer function f.
This step-by-step approach makes solving composite functions easier and ensures you don't mix up the order.
function evaluation
Function evaluation is the process of finding the output of a function given an input value. To evaluate a function, you substitute the input into the function's formula and compute the result.
This is a critical skill in solving composite functions. In our example, we begin by evaluating \(g(0)\):
  • For \(g(x) = 2x + 3\), substitute in x = 0:
  • \(g(0) = 2(0) + 3 = 3.\)
Next, we use this result to evaluate the outer function f with this new input:
  • For \(f(x) = x^2 + 4\), substitute in x = 3:
  • \(f(3) = 3^2 + 4 = 9 + 4 = 13.\)
So, \(f(3) = 13.\) Understanding these steps helps you accurately evaluate more complex expressions.
algebraic functions
Algebraic functions are functions built by combining basic operations like addition, subtraction, multiplication, division, and exponentiation.
In our provided functions, \(f, g,\) and \(h\), we see these operations clearly:
  • The function \(f(x) = x^2 + 4\) includes squaring and addition.
  • The function \(g(x) = 2x + 3\) involves multiplication and addition.
  • The function \(h(x) = x - 5\) involves subtraction.
Since these operations are fundamental to algebra, these functions are called algebraic functions.
Knowing how to manipulate and combine these basic operations is the foundation of working with more complex mathematical expressions. This becomes particularly essential when dealing with composite functions, as seen in our problem example.

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