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Magazine Chapter 2: Problem 69
Let \(f(x)=3 x^{2}-x\) and \(g(x)=4 x-2 .\) Find the following. $$f(x+1)$$
Short Answer
Expert verified
f(x+1) = 3x^{2} + 5x + 2
Step by step solution
01
Rewrite the Function with the New Input
Replace every occurrence of the variable 'x' in the function definition of \(f(x)\) with \(x+1\). The function \(f(x)\) is given by:\[ f(x) = 3x^{2} - x \] Thus, \( f(x+1) = 3(x+1)^{2} - (x+1) \).
02
Expand the Squared Term
Expand \((x+1)^{2}\) using the binomial theorem:\[ (x+1)^{2} = x^{2} + 2x + 1 \].
03
Substitute and Simplify
Substitute \(x^{2} + 2x + 1\) back into the function:\[ f(x+1) = 3(x^{2} + 2x + 1) - (x + 1) \] Now distribute and simplify: \[ f(x+1) = 3x^{2} + 6x + 3 - x - 1 \]
04
Combine Like Terms
Combine the like terms to get the final simplified form: \[ f(x+1) = 3x^{2} + 5x + 2 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Precalculus
Precalculus provides the necessary foundation for calculus. It's a bridge between algebra and calculus. In precalculus, you study functions, which are essential to understanding how numbers relate to each other. A function shows how one quantity changes when another one changes. In our exercise, we have two functions: \( f(x) = 3x^2 - x \) and \( g(x) = 4x - 2 \). Learning how to manipulate these functions properly prepares you for calculus concepts.
By practicing precalculus problems, such as finding \( f(x+1) \), you deepen your understanding of function composition, which is key in further mathematical studies.
Make sure to:
By practicing precalculus problems, such as finding \( f(x+1) \), you deepen your understanding of function composition, which is key in further mathematical studies.
Make sure to:
- Understand the definitions and types of functions (linear, quadratic, polynomial, etc.)
- Get comfortable with substituting and transforming functions
- Simplify expressions to their most basic form
Polynomial Functions
Polynomial functions are expressions that involve variables with non-negative integer exponents. They can be simple, like \( g(x) = 4x - 2 \), or more complex, like \( f(x) = 3x^2 - x \). Breaking down polynomial functions, you need to know about terms, coefficients, and exponents. Each term in a polynomial function can be thought of as a part of an overall equation that contributes to its shape.
In our exercise, \( f(x) \) is a quadratic polynomial because it has a term with \( x^2 \). Quadratic polynomials graph as parabolas. When finding \( f(x+1) \), you substitute \( x+1 \) wherever 'x' appears in the original function. Here鈥檚 a quick guide:
In our exercise, \( f(x) \) is a quadratic polynomial because it has a term with \( x^2 \). Quadratic polynomials graph as parabolas. When finding \( f(x+1) \), you substitute \( x+1 \) wherever 'x' appears in the original function. Here鈥檚 a quick guide:
- Identify each term in the polynomial
- Apply changes to each term one by one
- Combine like terms to simplify the expression
Binomial Theorem
The binomial theorem is a powerful tool for expanding expressions raised to a power. It's particularly useful when dealing with polynomial functions. The theorem states that: \[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \]. In our exercise, we use a simple case of the binomial theorem to expand \( (x+1)^2 \).
Here, \( (x+1)^2 = x^2 + 2x + 1 \). This form is achieved by applying the binomial coefficients and terms correctly. Remember to:
Here, \( (x+1)^2 = x^2 + 2x + 1 \). This form is achieved by applying the binomial coefficients and terms correctly. Remember to:
- Know the binomial coefficients (also called combinatorial numbers)
- Expand terms step-by-step
- Combine like terms carefully
Function Transformation
Function transformation involves altering a function鈥檚 formula to produce a new function. These changes can shift, stretch, compress, or flip the graph of the function. In the exercise, we perform a function transformation by finding \( f(x+1) \), essentially moving the entire function horizontally.
This involves a few key steps:
Understanding function transformation helps in visualizing how the graph of a function can be modified. It's pivotal for graphing complex functions and analyzing their behaviors.
This involves a few key steps:
- Substituting \( x+1 \) into the original function, replacing every instance of 'x'
- Simplifying the resulting expression to combine like terms
Understanding function transformation helps in visualizing how the graph of a function can be modified. It's pivotal for graphing complex functions and analyzing their behaviors.
