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Tarefa 1: Identificar as Funções Exponenciais

You are asked to circle the analytical expressions that define exponential functions. An exponential function is defined as $y = a^x$, where $a > 0$ and $a \neq 1$.

We analyze each expression:

1. $2^x$: Exponential function ✅
2. $\left(\frac{1}{3}\right)^x$: Exponential function ✅
3. $\frac{1}{x^3}$: Not exponential (this is a rational function).
4. $x^3$: Not exponential (this is a power function).
5. $1^x$: Not exponential (this is constant for $x$).
6. $4^{3x+1}$: Exponential function ✅
7. $5^{-x}$: Exponential function ✅ (can be written as $(1/5)^x$).
8. $x^2$: Not exponential (this is a polynomial function).
9. $\pi^{x+1}$: Exponential function ✅

Correct answers: $2^x$, $\left(\frac{1}{3}\right)^x$, $4^{3x+1}$, $5^{-x}$, $\pi^{x+1}$.

Tarefa 2: Estudo da Função $y = a^x$, com $a > 1$

1. Complete the Sentence

The sentence reads:
"As funções exponenciais têm um crescimento tanto mais rápido quanto maior for o valor de $a$ para a sua base."

Correct completion:
...quanto maior for o valor de $a$ para a sua base.
Larger bases lead to faster growth rates.

2. Domain, Zeros, and Monotonicity

For $y = a^x$, where $a > 1$ (e.g., $1.1^x, 1.5^x, 2^x, 3^x$):

• Domain: $(-\infty, \infty)$ (all real numbers).
• Zeros: None (the function never touches zero).
• Monotonicity: The functions are strictly increasing over the entire domain $(-\infty, \infty)$.

3. Asymptotes

Exponential functions $y = a^x$ with $a > 1$ have a horizontal asymptote at $y = 0$ as $x \to -\infty$.

4. Intersection with the $y$-axis

For all $y = a^x$, the intersection with the $y$-axis occurs when $x = 0$:
$y = a^0 = 1.$
Thus, the point of intersection is $(0, 1)$ for all functions.

Would you like a detailed explanation of any part or additional practice questions?

5 Related Questions

1. What are the key differences between exponential and polynomial functions?
2. How does changing the base $a$ in $y = a^x$ affect the graph?
3. Can $y = a^x$ ever have a negative value? Why or why not?
4. How do the graphs of $y = 2^x$ and $y = 2^{-x}$ compare?
5. Why is $y = 1^x$ considered constant instead of exponential?

Tip

When analyzing exponential graphs, always check their behavior as $x \to \infty$ and $x \to -\infty$; this helps identify asymptotes and growth trends.