Textbook Question
Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.
Find ƒ(g(h( x))).
0. Functions
Combining Functions
7:01 minutesProblem 93
Textbook Question
{Use of Tech} Polynomial calculations
Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)
ƒ(ƒ(x)) = 9x - 8
Verified step by step guidance1
Determine the degree of the polynomial \( f(x) \). Since \( f(f(x)) = 9x - 8 \) is a linear polynomial, assume \( f(x) \) is a linear polynomial of the form \( ax + b \).
Substitute \( f(x) = ax + b \) into \( f(f(x)) \) to get \( f(ax + b) = a(ax + b) + b = a^2x + ab + b \).
Set the expression \( a^2x + ab + b \) equal to \( 9x - 8 \) to find the coefficients. This gives the system of equations: \( a^2 = 9 \) and \( ab + b = -8 \).
Solve the equation \( a^2 = 9 \) to find possible values for \( a \). The solutions are \( a = 3 \) or \( a = -3 \).
Substitute each value of \( a \) into the equation \( ab + b = -8 \) to find the corresponding value of \( b \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Functions
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where n is a non-negative integer representing the degree of the polynomial. Understanding polynomial functions is crucial for manipulating and solving equations involving them.
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Composition of Functions
The composition of functions involves applying one function to the results of another. For two functions f and g, the composition is denoted as (f ∘ g)(x) = f(g(x)). In the context of the question, we need to find a polynomial f such that when it is composed with itself, it equals another polynomial, 9x - 8. This concept is essential for understanding how to manipulate and derive new functions from existing ones.
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Finding Coefficients
Finding coefficients in a polynomial involves determining the specific values that multiply the variable terms to satisfy given conditions or equations. In this case, after establishing the degree of the polynomial f, one must substitute a polynomial of that degree into the equation f(f(x)) = 9x - 8 and solve for the coefficients. This process is fundamental in polynomial algebra and is key to solving the problem presented.
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