Problem 33
Question
Find \([g \circ h](x)\) and \([h \circ g](x)\) $$ \begin{array}{l}{g(x)=x+1} \\ {h(x)=2 x^{2}-5 x+8}\end{array} $$
Step-by-Step Solution
Verified Answer
\([g \circ h](x) = 2x^2 - 5x + 9\) and \([h \circ g](x) = 2x^2 - x + 5\).
1Step 1: Understanding Functions
We are given two functions, \(g(x) = x + 1\) and \(h(x) = 2x^2 - 5x + 8\). The goal is to find \([g \circ h](x)\) and \([h \circ g](x)\). Function composition combines two functions such that the output of one function becomes the input of the other.
2Step 1: Find \([g \circ h](x)\)
The composition \([g \circ h](x)\) means applying \(h(x)\) first, then applying \(g(x)\) to the result. First, substitute \(h(x)\) in place of \(x\) in \(g(x)\):\[g(h(x)) = g(2x^2 - 5x + 8)\]Since \(g(x) = x + 1\), replace \(x\) with \(2x^2 - 5x + 8\):\[g(h(x)) = (2x^2 - 5x + 8) + 1\]Simplify the expression:\[g(h(x)) = 2x^2 - 5x + 9\].
3Step 2: Find \([h \circ g](x)\)
The composition \([h \circ g](x)\) means applying \(g(x)\) first, then applying \(h(x)\) to the result. First, substitute \(g(x)\) in place of \(x\) in \(h(x)\):\[h(g(x)) = h(x+1)\]Since \(h(x) = 2x^2 - 5x + 8\), replace \(x\) with \((x+1)\):\[h(g(x)) = 2(x + 1)^2 - 5(x + 1) + 8\]Expand and simplify:\[= 2(x^2 + 2x + 1) - 5x - 5 + 8\]\[= 2x^2 + 4x + 2 - 5x - 5 + 8\]Combine like terms:\[= 2x^2 - x + 5\].
Key Concepts
Algebra: The Language of MathematicsPolynomial Functions: Exploring Quadratic RelationshipsFunction Operations: Composition and Its Implications
Algebra: The Language of Mathematics
Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in mathematical expressions and equations. It allows us to create general formulas that describe mathematical relationships. One of the fundamental elements of algebra is the concept of a function, which essentially defines a relationship where each input has a unique output.
In algebra, functions like the ones in the given exercise, namely, \(g(x) = x + 1\) and \(h(x) = 2x^2 - 5x + 8\), are often analyzed to understand various relationships between different quantities. These functions become particularly interesting when we explore their composition, which is an operation involving functions just like addition is an operation involving numbers.
Understanding how to manipulate and combine these functions using algebraic techniques is crucial in many areas of higher mathematics and practical problem-solving scenarios. Mastery of these basic principles sets the stage for solving complex problems across a multitude of real-world domains.
In algebra, functions like the ones in the given exercise, namely, \(g(x) = x + 1\) and \(h(x) = 2x^2 - 5x + 8\), are often analyzed to understand various relationships between different quantities. These functions become particularly interesting when we explore their composition, which is an operation involving functions just like addition is an operation involving numbers.
Understanding how to manipulate and combine these functions using algebraic techniques is crucial in many areas of higher mathematics and practical problem-solving scenarios. Mastery of these basic principles sets the stage for solving complex problems across a multitude of real-world domains.
Polynomial Functions: Exploring Quadratic Relationships
Polynomial functions are expressions that involve variables raised to whole number exponents. These functions can represent a wide range of curves on a graph, from simple lines to complex parabolic shapes. In the context of the given exercise, the function \(h(x) = 2x^2 - 5x + 8\) is a quadratic polynomial function because its highest exponent is 2.
Quadratic functions are ubiquitous in algebra due to their frequent appearance in modeling various phenomena, such as the path of projectiles or the maximum area of a rectangle with a given perimeter. A polynomial function like this one can be analyzed to determine its key features, such as its vertex, axis of symmetry, and roots (where the function equals zero).
Polynomials can be manipulated by operations such as addition, subtraction, and, as seen in the exercise, function composition. Composing polynomial functions can reveal new insights and produce new polynomial expressions. This is achieved by substituting one polynomial into another, which requires careful attention to ensure accurate algebraic manipulation.
Quadratic functions are ubiquitous in algebra due to their frequent appearance in modeling various phenomena, such as the path of projectiles or the maximum area of a rectangle with a given perimeter. A polynomial function like this one can be analyzed to determine its key features, such as its vertex, axis of symmetry, and roots (where the function equals zero).
Polynomials can be manipulated by operations such as addition, subtraction, and, as seen in the exercise, function composition. Composing polynomial functions can reveal new insights and produce new polynomial expressions. This is achieved by substituting one polynomial into another, which requires careful attention to ensure accurate algebraic manipulation.
Function Operations: Composition and Its Implications
Function operations, including composition, are essential tools in algebra that allow for the transformation and manipulation of functions to explore more complex relationships. Function composition is a process where the output of one function becomes the input of another.
In the exercise, we see two examples of function composition. The first, \([g \circ h](x)\), involves plugging the polynomial \(h(x) = 2x^2 - 5x + 8\) into the linear function \(g(x) = x + 1\). This gives a new polynomial \(g(h(x)) = 2x^2 - 5x + 9\), which combines the elements of both original functions into a single expression.
Meanwhile, \([h \circ g](x)\) involves plugging \(g(x) = x + 1\) into \(h(x)\), leading to a slightly different result. This highlights how the order of function composition can drastically change the resulting expression, producing \(h(g(x)) = 2x^2 - x + 5\).
Understanding and performing these operations correctly is a vital skill in algebra. Function composition allows mathematicians and scientists to model systems where these layers of relationships are key to gaining insights, making predictions, or further refining mathematical processes.
In the exercise, we see two examples of function composition. The first, \([g \circ h](x)\), involves plugging the polynomial \(h(x) = 2x^2 - 5x + 8\) into the linear function \(g(x) = x + 1\). This gives a new polynomial \(g(h(x)) = 2x^2 - 5x + 9\), which combines the elements of both original functions into a single expression.
Meanwhile, \([h \circ g](x)\) involves plugging \(g(x) = x + 1\) into \(h(x)\), leading to a slightly different result. This highlights how the order of function composition can drastically change the resulting expression, producing \(h(g(x)) = 2x^2 - x + 5\).
Understanding and performing these operations correctly is a vital skill in algebra. Function composition allows mathematicians and scientists to model systems where these layers of relationships are key to gaining insights, making predictions, or further refining mathematical processes.
Other exercises in this chapter
Problem 33
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Determine whether each pair of functions are inverse functions. $$ \begin{array}{l}{g(x)=2 x+8} \\ {f(x)=\frac{1}{2} x-4}\end{array} $$
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For Exercises 33 and \(34,\) use the following information. A women's regulation-sized basketball is slightly smaller than a men's basketball. The radius \(r\)
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