A set of questions on the concept of a function in calculus is presented along with detailed solutions. These exercises are designed to help you gain a deep understanding of functions, including their domain and range. If you have difficulty with a question, review the related definitions and theorems.
True or False: Are the two functions \( f \) and \( g \) defined by \[ f(x) = 3x + 3, \quad x \in \mathbb{R}, \quad \text{and} \quad g(t) = 3t + 3, \quad t > 0 \] equal?
Answer: False. Two functions are equal if their rules are the same and their domains are identical.
If functions \( f \) and \( g \) have domains \( D_f \) and \( D_g \), respectively, then the domain of \( \frac{f}{g} \) is:
Answer: (C). Division by zero is not allowed.
True or False: The graph of \( f(x) \) and \( f(x+2) \) are the same.
Answer: False. The graph of \( f(x+2) \) is the graph of \( f(x) \) shifted 2 units to the left.
Let \( [a, b] \) be the domain of \( f \). What is the domain of \( f(x-3) \)?
Answer: (D). The graph of \( f(x-3) \) is shifted 3 units to the right, so the interval endpoints are increased by 3.
Let \( (a, +\infty) \) be the range of \( f \). What is the range of \( f(x)-4 \)?
Answer: (A). Since \( f(x) > a \), subtracting 4 gives \( f(x)-4 > a-4 \), so the range is \( (a-4, +\infty) \).
True or False: The equation \( y = |x| \), with \( y \ge 0 \), represents \( y \) as a function of \( x \).
Answer: True.
True or False: The equation \( x = |y| \), with \( x \ge 0 \), represents \( y \) as a function of \( x \).
Answer: False. Solving for \( y \) gives \( y = x \) or \( y = -x \); for one value of \( x \), there are two possible \( y \) values, so it is not a function.