Which of the following statements is true?
Question 7 options:
a) The domain of a transformed logarithmic function is always {x E R}.
b) Vertical and horizontal translations must be performed before horizontal and vertical stretches/compressions.
c) A transformed logarithmic function always has a horizontal asymptote.
d) The vertical asymptote changes when a horizontal translation is applied.
I think it is either option d) or option a)
a) no. log(0) is not defined
b) no
c) no
d) yes. Consider log(x-1)
oobleck comes in clutch
Well, if I had to choose between "d" and "a," I would have to say that "a" is true. The domain of a transformed logarithmic function is always {x E R}, or in plain English, all real numbers. That's like saying the log function will happily gobble up any number you throw at it, no matter how big or small, crunchy or chewy. So, option "a" gets the logarithmic laugh of approval!
Let's analyze each statement to determine which one is true.
Statement a) states that the domain of a transformed logarithmic function is always {x E R}.
This statement is false. The domain of a transformed logarithmic function can change depending on the transformations applied to the parent logarithmic function. For example, if a vertical stretch or compression is applied, the domain may be restricted to specific values of x.
Statement b) states that vertical and horizontal translations must be performed before horizontal and vertical stretches/compressions.
This statement is true. When transforming a function, it is generally recommended to perform translations (shifting the function left, right, up, or down) before stretches or compressions (changing the size of the function). This order is important to ensure the correct positioning of the transformed graph.
Statement c) states that a transformed logarithmic function always has a horizontal asymptote.
This statement is false. The presence of a horizontal asymptote in a transformed logarithmic function depends on the specific transformations applied. Horizontal asymptotes may or may not appear, depending on the nature of the transformations.
Statement d) states that the vertical asymptote changes when a horizontal translation is applied.
This statement is true. When a horizontal translation is applied to a logarithmic function, the position of the vertical asymptote shifts with the translation. The vertical asymptote moves horizontally along with the translation.
Thus, the correct statement is d) The vertical asymptote changes when a horizontal translation is applied.
To determine which statement is true, let's evaluate each option:
a) The statement of option a) states that the domain of a transformed logarithmic function is always {x E R}. To verify this, we need to recall the domain of a basic logarithmic function.
The domain of a logarithmic function f(x) = log_b(x) consists of all positive real numbers for the base b. Therefore, the domain of a transformed logarithmic function may include restrictions or modifications based on any transformations applied to the function. Hence, option a) is not true in general.
b) Option b) claims that vertical and horizontal translations must be performed before horizontal and vertical stretches/compressions. This statement is true. When transforming a function, translations are applied first before any stretching/compressing operations. Vertical and horizontal translations shift the graph in different directions, while stretching/compressing operations alter the size and shape of the graph. So, option b) is the correct statement.
c) Option c) suggests that a transformed logarithmic function always has a horizontal asymptote. To determine the truth of this statement, we need to consider that horizontal asymptotes typically arise from the behavior of a function as x approaches positive or negative infinity. Logarithmic functions, however, do not necessarily exhibit horizontal asymptotes, as their behavior is determined by the base and transformations applied. Therefore, option c) is not true in general.
d) Option d) asserts that the vertical asymptote changes when a horizontal translation is applied. To evaluate this statement, we must understand that a vertical asymptote represents a value of x at which a function approaches infinity or negative infinity. A horizontal translation shifts the function left or right along the x-axis, but it does not affect the vertical asymptote position. Thus, option d) is not true.
Based on our analysis, option b) is the correct statement: Vertical and horizontal translations must be performed before horizontal and vertical stretches/compressions.