Question

Consider the function f(x) = log(x). Write a function rule for g(x) which is a reflection of f(x) over the x-axis, then a translation 3 units down and 7 units right.

Answer 1

The resultant function rule for g(x) after reflecting f(x) = log(x) over the x-axis, then translating 3 units down and 7 units right is g(x) = -log(x - 7) - 3.

Step-by-step explanation:

The student is asking us to perform a series of transformations on the logarithmic function f(x) = log(x) to obtain a new function g(x). To reflect the function f(x) over the x-axis, we take the negative of the function, which gives us -log(x). After reflecting f(x) over the x-axis to get -log(x), we then translate it 3 units down and 7 units to the right. Translating the function down means subtracting 3 from it, resulting in -log(x) - 3. Finally, moving the function 7 units to the right is achieved by replacing x with x - 7, thus we get -log(x - 7) - 3 as the final function rule for g(x).

Expressing the full transformation, we have:

• Reflection over the x-axis: f(x) becomes -f(x)
• Translation 3 units down: -f(x) becomes -f(x) - 3
• Translation 7 units right: -f(x) - 3 becomes -log(x - 7) - 3

The resultant function g(x) is -log(x - 7) - 3, which is the correct option for the function rule after the described transformations.