Question

the graph of the exponential function f(x) = 3^x is shown.
domain:
inequality:
set notation:
interval notation:
range:
inequality
set notation:
interval notation:
end behavior

Explanation:

Step1: Determine Domain (x - values)

For exponential function \( f(x) = 3^x \), there are no restrictions on \( x \). The graph extends infinitely left and right.
Inequality: \( x \in \mathbb{R} \) (or \( -\infty < x < \infty \))
Set Notation: \( \{ x | x \in \mathbb{R} \} \)
Interval Notation: \( (-\infty, \infty) \)

Step2: Determine Range (y - values)

Exponential function \( 3^x \) has \( y>0 \) (since \( 3^x \) is always positive, and as \( x \to -\infty \), \( 3^x \to 0 \) but never reaches 0; as \( x \to \infty \), \( 3^x \to \infty \)).
Inequality: \( y > 0 \)
Set Notation: \( \{ y | y > 0, y \in \mathbb{R} \} \)
Interval Notation: \( (0, \infty) \)

Step3: End Behavior

As \( x \to -\infty \), \( 3^x \to 0 \) (since the graph approaches the x - axis from above as \( x \) becomes very negative). As \( x \to \infty \), \( 3^x \to \infty \) (the graph rises rapidly as \( x \) becomes very positive).
So, as \( x \to -\infty \), \( f(x) \to 0 \); as \( x \to \infty \), \( f(x) \to \infty \)

Answer:

Domain

• Inequality: \( -\infty < x < \infty \)
• Set Notation: \( \{x\mid x\in\mathbb{R}\} \)
• Interval Notation: \( (-\infty, \infty) \)

Range

• Inequality: \( y > 0 \)
• Set Notation: \( \{y\mid y > 0,y\in\mathbb{R}\} \)
• Interval Notation: \( (0, \infty) \)

End Behavior

As \( x\to-\infty \), \( f(x)\to 0 \); as \( x\to\infty \), \( f(x)\to\infty \)