Q: If $$f(x) is $$defined by a graph with a removable discontinuity at $$x = 2$$, which of the following is true?

The limit as $$x \to 2$$ exists but $$f(2) is $$not equal to the limit.

Question 1

Given the quadratic function $$f(x) = -x^2$ + 4$$, what is the domain and range of $$f(x)$$?

Accepted Answer

Domain: $$(-\infty, \infty)$$, Range: $$(-\infty, 4]$$

Question 2

Which of the following statements is true about the function $$f(x) = -x^2$ + 4$$?

Accepted Answer

It is a function but not invertible over its entire domain.

Question 3

Given the function $$f(x) = x^3$, what is the inverse function f$$^{-1}$$(x)$$?

Accepted Answer

$$f^{-1}(x$$) = \sqrt[3]{x}$$

Question 4

If $$y = x^3$, what is $$\frac{dy}{dx}$$ and $$\frac{dx}{dy}$$ at x = 2$?

Accepted Answer

$$\frac{dy}{dx} = 3x^2$ = 12$$, $$\frac{dx}{dy} = \frac{1}{12}$$

Question 5

Consider the function $$f(x)$$ defined by the graph with jumps and holes. If $$\lim_{x 	o -10} f(x) = 2$$ and $$f(-10) = 3$$, is $$f(x)$$ continuous at $$x = -10$$?

Accepted Answer

No, because the limit exists but does not equal the function value.

Question 6

Given $$f(x) = \frac{1}{x}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$(-\infty, 0) \cup (0, \infty)$$

Question 7

For the function $$f(x) = x^3$, which of the following is true about its graph?

Accepted Answer

It is symmetric about the origin.

Question 8

Given the function $$f(x)$$ defined by the graph with a jump at $$x = -27$$, what is $$\lim_{x 	o -27^-} f(x)$$ and $$\lim_{x 	o -27^+} f(x) if $$the left and right limits are not equal?

Accepted Answer

The left and right limits are not equal, so the limit does not exist at $$x = -27.$$

Question 9

If $$f(x) is $$continuous at $$x = a$$, which of the following must be true?

Accepted Answer

$$\lim_{x 	o a} f(x) = f(a)$$

Question 10

Given $$f(x) = x^3$ and its inverse f$$^{-1}$$(x) = \sqrt[3]{x}$$, what is the relationship between their derivatives at corresponding points?

Accepted Answer

$$f'(x) \cdot \left( f$$^{-1}$$ \right)'(f(x)) = 1$$

Question 11

A function $$f(x)$$ passes the vertical line test. What does this imply?

Accepted Answer

For every $$x in $$the domain, there is exactly one $$y$$ value.

Question 12

Given the function $$f(x) = x^3$, what is the range of f(x$$)$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 13

If $$f(x) is $$defined by a graph with a removable discontinuity at $$x = 2$$, which of the following is true?

Accepted Answer

The limit as $$x 	o 2$$ exists but $$f(2) is $$not equal to the limit.

Question 14

Given $$f(x) = x^3$ and g(x$$) = \sqrt[3]{x}$$, which statement is true?

Accepted Answer

$$g(x) is $$the inverse of $$f(x).$$

Question 15

Suppose $$f(x) is a $$function with domain $$(-\infty, \infty)$$ and range $$(-\infty, 4]. $$Which of the following could be the graph of $$f(x)$$?

Accepted Answer

A downward opening parabola with vertex at $$(0, 4)$$

Question 16

If $$f(x) is $$invertible, which of the following must be true?

Accepted Answer

It passes the horizontal line test.

Question 17

Given $$f(x) = x^3$ and f$$^{-1}$$(x) = \sqrt[3]{x}$$, what is $$f(f^{-1}(x$$))$$?

Accepted Answer

$$x$$

Question 18

If $$f(x) is $$not continuous at $$x = a$$, which of the following could be true?

Accepted Answer

$$\lim_{x 	o a} f(x)$$ does not exist.

Functions, Domains, Ranges, and Inverses: Calculus Study Notes

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