The problem asks you to graph the exponential function $g(x) = \left( \frac{4}{5} \right)^x$.

Key Characteristics of the Function:

1. Base $b = \frac{4}{5}$: Since $\frac{4}{5} < 1$, this is a decaying exponential function.
2. Behavior: As $x \to \infty$, $g(x) \to 0$ (approaches zero). As $x \to -\infty$, $g(x) \to \infty$ (grows larger).

Step-by-Step Solution:

To graph this, you need to calculate a few points by substituting values of $x$. Let’s calculate the values:

Example Points:

1. When $x = 0$: $g(0) = \left( \frac{4}{5} \right)^0 = 1$ So, $(0, 1)$.

2. When $x = 1$: $g(1) = \frac{4}{5} = 0.8$ So, $(1, 0.8)$.

3. When $x = -1$: $g(-1) = \left( \frac{4}{5} \right)^{-1} = \frac{5}{4} = 1.25$ So, $(-1, 1.25)$.

4. When $x = 2$: $g(2) = \left( \frac{4}{5} \right)^2 = \frac{16}{25} = 0.64$ So, $(2, 0.64)$.

5. When $x = -2$: $g(-2) = \left( \frac{4}{5} \right)^{-2} = \frac{25}{16} \approx 1.56$ So, $(-2, 1.56)$.

Points to Plot:

• $(0, 1)$
• $(1, 0.8)$
• $(-1, 1.25)$
• $(2, 0.64)$
• $(-2, 1.56)$

Graphing Tips:

1. Plot these points on the graph.
2. Connect the points with a smooth curve that decreases as $x$ increases.
3. Ensure the curve approaches zero on the right ($x \to \infty$) and increases on the left ($x \to -\infty$).

Would you like further assistance plotting the graph or understanding any part?

Related Questions:

1. What happens if the base $b$ is greater than 1?
2. How do you determine the horizontal asymptote of this function?
3. How would you shift this graph vertically or horizontally?
4. How do transformations affect the exponential decay rate?
5. How can this function model real-world decay scenarios?

Tip: Always check the behavior at $x = 0$ and a few positive and negative $x$-values to ensure accuracy in plotting exponential graphs!