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Chapter 5: Problem 43

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

Short Answer

Expert verified
(g \circ f)(x) = 2x^2 + 11

Step by step solution

01

Understand Composition of Functions

The expression \(g \circ f\)(x) means we need to find \(g(f(x))\). This involves substituting the output of function \(f(x)\) into the function \(g(x)\).
02

Compute f(x)

The function \(f(x)\) is given as \(f(x) = x^2 + 4\).
03

Substitute f(x) into g(x)

Next, use the result of \(f(x)\) as input for \(g(x)\). The function \(g(x)\) is given as \(g(x) = 2x + 3\). Substitute \(f(x)\) into \(g\): \(g(f(x)) = g(x^2 + 4)\).
04

Simplify the Expression

Finally, replace \(x\) in \(g(x) = 2x + 3\) with \(x^2 + 4\) to get \(g(x^2 + 4) = 2(x^2 + 4) + 3\). Simplifying, this yields \(2x^2 + 8 + 3 = 2x^2 + 11\).

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Key Concepts

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

function notation
Function notation is a concise way to represent a function mathematically using symbols. For example, instead of writing 'y equals x squared plus 4', we use function notation:
  • Given: \( f(x) = x^2 + 4 \).
    • Here, \( f \) is the name of the function. \( x \) is the variable or input symbol, and \( x^2 + 4 \) is the function rule.Function notation helps in clearly specifying what happens to the variable as it goes through the function. When performing function operations like compositional functions, it is crucial to understand and correctly apply function notation. This way, we can substitute one function into another efficiently.
substitution
Substitution involves replacing a variable with a given value or another expression. In composition of functions, like finding \( g(f(x)) \), substitution plays a key role.Consider the function \( f(x) = x^2 + 4 \). To compose it with \( g(x) = 2x + 3 \), follow these steps:
  • Compute \( f(x) \): \( f(x) = x^2 + 4 \).
  • Next, substitute \( f(x) \) into \( g(x) \). This means you replace \( x \) in \( g(x) \) with \( f(x) \).
  • So, we get \( g(f(x)) = g(x^2 + 4) \).
Now, use the definition of \( g(x) \) to substitute: \( g(x^2 + 4) = 2(x^2 + 4) + 3 \). This kind of substitution keeps the logical flow and applies the rules of the function properly.
simplification in algebra
Simplification in algebra is about reducing expressions to their simplest form. This makes them easier to understand and use in further calculations.Using our previous example: We have the expression \[ g(x^2 + 4) = 2(x^2 + 4) + 3\]. Now, let's simplify it step by step:
  • First, distribute the 2 across the terms inside the parentheses: \( 2 \cdot x^2 + 2 \cdot 4 \).
  • This gives \( 2x^2 + 8 \).
  • Don't forget to add the last term: \( g(x^2 + 4) = 2x^2 + 8 + 3 \).
Combining the constant terms, we get our simplified result: \( 2x^2 + 11 \).Simplification makes the expression easy to handle and helps in understanding the underlying relationships between variables.

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Most popular questions from this chapter

Write each number in standard notation. $$ 5.42 \times 10^{-4} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \frac{2\left(-m^{-1}\right)^{-4}}{9\left(m^{-3}\right)^{2}} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{\left(2 m^{3} x^{2}\right)^{-1}\left(3 m^{4} x\right)^{-3}}{\left(m^{2} x^{3}\right)^{3}\left(m^{2} x\right)^{-5}} $$

Find each product. \(6 m n^{3}\left(3 m^{4} n\right)\)

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ 4^{4} \cdot 4^{-6} $$
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