Given that f(x) = (4 - 7x) / (6 - x), find f(-2).
Understand the Problem
The question asks to find the value of the function f(x) when x = -2, given that f(x) = (4 - 7x) / (6 - x). This involves substituting -2 for x in the given function and simplifying the expression to find the numerical value of f(-2).
Answer
$f(-2) = \frac{9}{4}$
Answer for screen readers
$f(-2) = \frac{9}{4}$
Steps to Solve
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Substitute $x = -2$ into the function Given the function $f(x) = \frac{4 - 7x}{6 - x}$, substitute $-2$ for $x$: $$f(-2) = \frac{4 - 7(-2)}{6 - (-2)}$$
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Simplify the numerator Simplify the expression in the numerator: $$4 - 7(-2) = 4 + 14 = 18$$
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Simplify the denominator Simplify the expression in the denominator: $$6 - (-2) = 6 + 2 = 8$$
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Write the simplified fraction Substitute the simplified numerator and denominator into the fraction: $$f(-2) = \frac{18}{8}$$
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Simplify the fraction Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2: $$f(-2) = \frac{18 \div 2}{8 \div 2} = \frac{9}{4}$$
$f(-2) = \frac{9}{4}$
More Information
The value of the function $f(x) = \frac{4 - 7x}{6 - x}$ when $x = -2$ is $\frac{9}{4}$. This can also be expressed as the mixed number $2\frac{1}{4}$ or the decimal $2.25$.
Tips
A common mistake is to incorrectly handle the negative signs when substituting $x = -2$ into the function. Specifically, errors can occur when calculating $-7(-2)$ in the numerator or $6 - (-2)$ in the denominator. Paying careful attention to the order of operations and the rules for multiplying and subtracting negative numbers can prevent these errors.
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