tDan paid the grocery bill of $76.45 with $100. Mistakenly the cashier gave Dan only $13.55 in change. How much more does the ca
1 answer:
You might be interested in
Answer:
If the expression is , then the answer is the first option.
If the expression is , then the answer is the third option.
Step-by-step explanation:
Remember that when you have a radical expression in the form , you can rewrite as:
Then:
- If the expression is , then you can rewrite it in the following radical form:
(This form matches with the first option.)
- If the expression is , then you can rewrite it in the following radical form:
(This form matches with the third option).
Note that f(x) as given is <em>not</em> invertible. By definition of inverse function,
which is a cubic polynomial in with three distinct roots, so we could have three possible inverses, each valid over a subset of the domain of f(x).
Choose one of these inverses by restricting the domain of f(x) accordingly. Since a polynomial is monotonic between its extrema, we can determine where f(x) has its critical/turning points, then split the real line at these points.
f'(x) = 3x² - 1 = 0 ⇒ x = ±1/√3
So, we have three subsets over which f(x) can be considered invertible.
• (-∞, -1/√3)
• (-1/√3, 1/√3)
• (1/√3, ∞)
By the inverse function theorem,
where f(a) = b.
Solve f(x) = 2 for x :
x³ - x + 2 = 2
x³ - x = 0
x (x² - 1) = 0
x (x - 1) (x + 1) = 0
x = 0 or x = 1 or x = -1
Then can be one of
• 1/f'(-1) = 1/2, if we restrict to (-∞, -1/√3);
• 1/f'(0) = -1, if we restrict to (-1/√3, 1/√3); or
• 1/f'(1) = 1/2, if we restrict to (1/√3, ∞)