In both functions, the base of the exponent is $0.3$, which is between $0$ and $1$. For such a base, does the function get bigger when the exponent gets bigger or when the exponent gets smaller?

For each exponent expression ($x^2+1$ and $x^2-2$, or the simplified version), find its smallest possible value when $x \ge 0$. Then plug that $x$ into the function to get its maximum value and compare it to the constants or coefficients.

In function II, try rewriting $0.09$ in terms of $0.3$. How can you express $0.09$ as a power of $0.3$, and what happens if you then combine the exponents?

Type y1 = 20*(0.3)^(x^2+1) {x>=0} so Desmos only shows the graph for $x \ge 0$. Then use the graph or a table (click on the function and choose “Table”) to see where $y_1$ is largest and what that maximum $y$-value is.

Type y2 = 20*(0.09)*(0.3)^(x^2-2) {x>=0}. If you want to see the simplification, you can also enter y3 = 20*(0.3)^(x^2) {x>=0} and notice that the two graphs overlap exactly, confirming the simplification.

Use the table or trace along each graph for $x \ge 0$ to find the highest $y$-value each function reaches (you will see this at $x=0$). Then check whether that $y$-value equals any constant or coefficient that appears in that function’s equation, and decide which function fits the condition described in the question.

The question asks which function has its maximum value shown explicitly in the equation as a constant or coefficient.

For each function, you need to:

• Find its maximum value for $x \ge 0$.
• Check whether that maximum value is one of the numbers written in the formula (like $20$, $0.3$, $0.09$, etc.).

In both functions, the base of the exponential is $0.3$, which is between $0$ and $1$.

For a number $b$ where $0<b<1$:

• As the exponent increases, $b^\text{(exponent)}$ decreases.
• So the function is largest when the exponent is as small as possible.

Here the exponents are expressions involving $x^2$, so we want to find the minimum value of those exponent expressions for $x \ge 0$.

Function I is

For $x \ge 0$, $x^2 \ge 0$, so the exponent $x^2+1$ is smallest when $x=0$:

• At $x=0$, the exponent is $0^2+1 = 1$.
• So

Because the exponent $x^2+1$ only gets larger as $x$ moves away from $0$, and the base $0.3$ is less than $1$, $f(0)=6$ is the maximum value of $f(x)$ for $x \ge 0$.

Now compare: the constants or coefficients in $f(x)$ are $20$, $0.3$, and $1$ (in the exponent), but none of these equals $6$, the maximum value.

Function II is

Notice that $0.09 = 0.3^2$, so we can rewrite:

so

For $x \ge 0$, $x^2 \ge 0$, and the exponent $x^2$ is smallest when $x=0$.

• At $x=0$, the exponent is $0$, so

As $x$ moves away from $0$, $x^2$ increases, so $(0.3)^{x^2}$ decreases, meaning $g(0)=20$ is the maximum value of $g(x)$ for $x \ge 0$.

Here the maximum value, $20$, does appear as a coefficient in the equation. Combining this with the result for function I, only function II satisfies the condition, so the correct choice is II only.