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Chapter 1: Problem 1

If \(f(x)=3 x^{4}-5 x^{2}+9\), find \(f(x-1)\)

Short Answer

Expert verified
The function \(f(x-1)\) simplifies to \(f(x - 1) = 3x^{4} - 12x^{3} + 21x^{2} - 14x + 13\).

Step by step solution

01

Understand the function

The given function is \(f(x)=3 x^{4}-5 x^{2}+9\). We need to find the value of the function at \(x - 1\), which means we need to substitute \(x - 1\) into the function and simplify.
02

Substitute \(x - 1\) into the function

Replace \(x\) with \(x - 1\) in the given function. So, \(f(x - 1) = 3(x - 1)^{4} - 5(x - 1)^{2} + 9\)
03

Simplify the expression

Expand the power of 4 and power of 2 expressions and then simplify the equation to get the final form of the function. After simplifying, we get \(f(x - 1) = 3x^{4} - 12x^{3} + 21x^{2} - 14x + 13\)

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Key Concepts

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

Polynomial Functions
Polynomial functions are a fundamental component of algebra and calculus. They consist of variables and coefficients that involve only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In the expression given in the exercise, \( f(x) = 3x^4 - 5x^2 + 9 \), you can see the variable \(x\) is raised to different powers, specifically 4 and 2.
  • Identifying terms: Each segment like \(3x^4\) or \(-5x^2\) is known as a 'term'.
  • Coefficients: These are the numerical parts of the terms; in \(3x^4\), 3 is the coefficient.
  • Degree of the polynomial: This is determined by the highest power to which the variable is raised. In \(f(x)\), the degree is 4.

Understanding polynomial functions is vital because they model a variety of real-world scenarios and lay the groundwork for more complex mathematical concepts.
Function Notation
Function notation is a way to represent functions in a more structured form, helping to distinguish between input and output. The notation \( f(x) \) is read as "\( f \) of \( x \)", where \( f \) is the name of the function, and \( x \) is the input value.
  • Function representation: The equation \( f(x) = 3x^4 - 5x^2 + 9 \) tells us how \( x \) values are turned into \( f(x) \) values.
  • Changing inputs: You can substitute different values into \( x \) to see how the output changes.
  • Specific values: If you want \( f \) at a specific \( x \), such as \( x=2 \), then you calculate \( f(2) \).

This notation is essential to express how changes to the input of a function affect the resulting output, and it's widely used across different branches of mathematics.
Mathematical Substitution
Mathematical substitution involves replacing one element of an equation with another. In the given problem, you replaced \( x \) with \( x-1 \) in the original function. This process allows for evaluating functions at different points or transforming them.
  • Evaluating derivatives: Substitution is crucial in calculus to evaluate derivatives and integrals.
  • Transforming and solving equations: It helps in simplifying complex expressions to solve equations more easily.
  • Performing function transformations: This is where you alter input to see how the function changes, very useful in graphing.

Substitution is a powerful tool for transforming equations into more workable forms or evaluating specific function outputs like \( f(x-1) \). Through practice, the concept becomes an intuitive skill that opens up a world of mathematical potential.

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Most popular questions from this chapter

Find the period of \(f(x)=3 \sin \\{2 x\\}+2 \cos \\{3 x\\}\)

If \(f(x)=a x+b\) and \(f(f(f(x)))=27 x+26\) where \(a\), \(b \in R\), find the value of \(a^{2}+b^{2}+2\)

If \(f(x)=\frac{x}{\sqrt{1+x^{2}}}\), prove that \(f(f(f(x)))=\frac{x}{\sqrt{1+3 x^{2}}}\)

Let \(f(x)=x+e^{x}+\sin x\) be defined on \([0,2]\). Then find its even and odd extension.

If \(g\) is the inverse of \(f\) and \(f^{\prime}(x)=\sin x\), find \(g^{\prime}(x)\).
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