Let f(x)=x2−4x−2f\left(x\right)=\frac{$$x^2-4$$}{x-2}f(x)=x−2x2−4 . Calculate f(x)f\left(x\right)f(x) for each value of xxx in the following table.

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

문제 7a

2장, 문제 7a

Let $f$\left$(x$\right$)=$\frac{x^2-4}{x-2}$$ . <IMAGE>
Calculate $f$\left$(x$\right$)$ for each value of $x$ in the following table.

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First, let's simplify the function $ f(x) = \frac{x^2 - 4}{x - 2} $. Notice that the numerator $ x^2 - 4 $ can be factored as a difference of squares: $ (x - 2)(x + 2) $.

Rewrite the function using the factored form: $ f(x) = \frac{(x - 2)(x + 2)}{x - 2} $.

Observe that the $ x - 2 $ terms in the numerator and the denominator can be canceled out, but only for $ x \neq 2 $ to avoid division by zero. Thus, $ f(x) = x + 2 $ for $ x \neq 2 $.

Now, for each value of $ x $ in the table, substitute $ x $ into the simplified function $ f(x) = x + 2 $ to find the corresponding value of $ f(x) $.

Remember to handle the case where $ x = 2 $ separately, as the original function is undefined at this point due to division by zero. Consider the limit as $ x $ approaches 2 if needed.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In this case, the function f(x) = (x^2 - 4) / (x - 2) is a rational function where the numerator is a polynomial of degree 2 and the denominator is a polynomial of degree 1. Understanding rational functions is crucial for analyzing their behavior, including identifying points of discontinuity and simplifying expressions.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing the expression to its simplest form by factoring and canceling common factors in the numerator and denominator. For the function f(x) = (x^2 - 4) / (x - 2), recognizing that the numerator can be factored as (x - 2)(x + 2) allows us to simplify the expression, which is essential for evaluating the function at specific values of x.

Evaluating Functions

Evaluating a function means substituting a specific value for the variable and calculating the output. In this context, evaluating f(x) for various values of x requires substituting those values into the simplified form of the function. This process is fundamental in calculus as it helps in understanding the function's behavior and its graphical representation.

Let f(x)=x2−4x−2f\(\left\)(x\(\right\))=\(\frac{x^2-4}{x-2}\)f(x)=x−2x2−4 . <IMAGE>Calculate f(x)f\(\left\)(x\(\right\))f(x) for each value of xxx in the following table.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Rational Functions

Simplifying Rational Expressions

Evaluating Functions

Let $f\(\left\)(x\(\right\))=\(\frac{x^2-4}{x-2}\)$ . <IMAGE>
Calculate $f\(\left\)(x\(\right\))$ for each value of $x$ in the following table.