To determine the transformations needed to form the function $g(x) = 1 - 10^{-x}$ from $f(x) = 10^x$, let’s analyze the transformations step-by-step.

1. Reflect about the y-axis:

   • To get $10^{-x}$ from $10^x$, we need to reflect $f(x) = 10^x$ about the y-axis. This transformation changes $x$ to $-x$.

2. Negate the function:

   • After obtaining $10^{-x}$, we need $-10^{-x}$, which requires us to negate the function.

3. Add 1 to the function:

   • Finally, to get $1 - 10^{-x}$, we add 1 to the transformed function.

Answers to the Questions

For the first question:

• The correct transformations of $f(x) = 10^x$ to obtain $g(x) = 1 - 10^{-x}$ are:
  • $-f(x)$ (negation)
  • $f(-x)$ (reflection about the y-axis)

For the second question:

• The correct transformations to the graph of $f(x) = 10^x$ that yield the graph of $g(x) = 1 - 10^{-x}$ are:
  • Reflect about the y-axis
  • Reflect about the x-axis (due to negating $f(x)$)

Would you like more details on any part of this process? Here are some related questions to deepen understanding:

1. What is the general process for reflecting a function about the y-axis?
2. How does negating a function affect its graph?
3. Can you provide examples of transformations that involve shifting instead of reflecting?
4. How would the transformation change if the function was $g(x) = 1 + 10^{-x}$?
5. What does each transformation look like graphically?

Tip: Always apply transformations in steps, starting with reflections and negations before shifts and translations for easier visualization.