The function g is defined by \(\mathrm{g(x) = 20 - 9(1/3)^{x+1}}\). What is the y-intercept of the graph of \(\mathrm{y...

1. TRANSLATE the problem requirements

• Given: \(\mathrm{g(x) = 20 - 9(1/3)^{(x+1)}}\)
• Find: The y-intercept of the graph

• Key insight: The y-intercept occurs where the graph crosses the y-axis, which is when \(\mathrm{x = 0}\)

2. INFER the solution approach

• To find the y-intercept, I need to evaluate \(\mathrm{g(0)}\)
• This will give me the y-coordinate when \(\mathrm{x = 0}\), forming the point \(\mathrm{(0, y)}\)

3. SIMPLIFY by substituting x = 0

• \(\mathrm{g(0) = 20 - 9(1/3)^{(0+1)}}\)
• \(\mathrm{g(0) = 20 - 9(1/3)^1}\)
• Since any number to the first power equals itself: \(\mathrm{g(0) = 20 - 9(1/3)}\)
• \(\mathrm{g(0) = 20 - 3}\)
• \(\mathrm{g(0) = 17}\)

4. APPLY CONSTRAINTS to express the final answer

• The y-intercept is the point \(\mathrm{(0, 17)}\)

Answer: C. (0, 17)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might not understand what "y-intercept" means mathematically. They may think it refers to some special property of the function rather than simply the point where \(\mathrm{x = 0}\). This confusion leads them to attempt complex manipulations of the exponential function instead of the straightforward substitution. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly substitute \(\mathrm{x = 0}\) but make arithmetic errors. A common mistake is computing \(\mathrm{(1/3)^1}\) incorrectly, perhaps thinking it equals \(\mathrm{1/9}\) or getting confused with fraction arithmetic when calculating \(\mathrm{20 - 3}\). This computational error may lead them to select Choice B (0, 11) or another incorrect option.

The Bottom Line:

This problem tests whether students understand the fundamental definition of y-intercept and can execute basic substitution with exponential expressions. The exponential function looks intimidating, but the actual calculation is straightforward once students recognize they simply need to plug in \(\mathrm{x = 0}\).