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Chapter 5: Problem 40

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

Short Answer

Expert verified
The value of \((h \circ g)(-2)\) is \-6\.

Step by step solution

01

Understand the composition of functions

The composition of functions \((h \circ g)(x)\) means we need to substitute \g(x)\ into \h(x)\. It can be written as \(h(g(x))\).
02

Evaluate \g(x)\ at \-2\

Calculate \g(-2)\ using the function \g(x) = 2x + 3\. Substitute \-2\ for \x\: \[g(-2) = 2(-2) + 3 = -4 + 3 = -1\]
03

Substitute the result into \h(x)\

Now use the result from Step 2, which is \-1\, and substitute it into the function \h(x) = x - 5\. Calculate \[h(-1) = -1 - 5 = -6\]

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Key Concepts

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

Function Evaluation
Function evaluation involves finding the value of a function for a particular input.
In this exercise, we have three functions:
  • h(x) = x - 5
  • g(x) = 2x + 3
  • f(x) = x^2 + 4
Evaluating these functions means substituting specific values for x and calculating the result. For example, if we want to find the value of g(x) when x = -2, we substitute -2 into the equation for g(x), giving us 2(-2) + 3.
The process is just plug-and-play.
As seen in the solution, this gives us -4 + 3, which is -1.
Substitution in Mathematics
Substitution is replacing a variable with a specific value or another expression.
In the context of function composition, substitution is essential. Here, we compose the functions h and g, resulting in \( h \circ g \).
This notation means we first evaluate g(x) and then use that result as the input for h(x). For (h \circ g)(-2), we begin by evaluating g(-2) and find it equals -1.
Finally, we substitute -1 into h(x), yielding h(-1), which results in -6. Understanding substitution helps with breaking down complex problems into manageable steps.
Intermediate Algebra
Intermediate Algebra focuses on understanding and manipulating algebraic expressions and equations.
Concepts including function evaluation and composition, like in this problem, are crucial topics.
Mastery often requires practice with manipulating different mathematical expressions and performing operations step-by-step. In this exercise, we use algebraic manipulation to substitute one function into another. We also perform basic arithmetic operations鈥攁ddition, subtraction, and multiplication鈥攄emonstrating foundational skills in algebra.
Skills learned in Intermediate Algebra are building blocks for higher-level mathematics and problem-solving in various disciplines.

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Most popular questions from this chapter

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \frac{(-8 x y) y^{3}}{4 x^{5} y^{4}} $$

Write each number in scientific notation. $$ 0.830 $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ a^{-3} a^{2} a^{-4} $$

Write each number in scientific notation. $$ 0.00000692 $$

Find each product. $$(5 r-s)^{3}$$
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