An exponential function is defined for all x!

\(f(x) = a^x\) (where a > 0, a ≠ 1)

Domain: ℝ (all real numbers)

💡 Why?

You can raise a positive base to any power — positive, negative, fraction, zero — it all works!

For example: \(2^3 = 8\), \(2^{-1} = 0.5\), \(2^0 = 1\), \(2^{0.5} = \sqrt{2}\)

Example 1: \(f(x) = 2^x\)

Positive base (2 > 0) → defined for all x

Domain: ℝ

Example 2: \(f(x) = e^x\)

e ≈ 2.718 is a positive base → defined for all x

Domain: ℝ

Example 3: \(f(x) = 3^{x+1}\)

The exponent can be anything → defined for all x

Domain: ℝ

Example 4: \(f(x) = 5 \cdot 2^{3x-7} + 4\)

Operations on an exponential function do not change the domain

Domain: ℝ

Example 5: \(f(x) = \left(\frac{1}{2}\right)^x\)

The base ½ is positive → defined for all x

Domain: ℝ

💡 Important reminder:

The range of an exponential function \(a^x\) (with a > 0) is (0, ∞) — always positive!

So you can always take the square root of an exponential function!

Example 6: \(f(x) = \sqrt{2^x}\)

Condition: \(2^x \geq 0\)

Check: \(2^x > 0\) always! (exponential is always positive)

Domain: ℝ

Example 7: \(f(x) = \sqrt{e^x - 1}\)

Condition: \(e^x - 1 \geq 0\)

\(e^x \geq 1\)

\(x \geq \ln(1) = 0\)

Domain: \(x \geq 0\)

Example 8: \(f(x) = \sqrt{4 - 2^x}\)

Condition: \(4 - 2^x \geq 0\)

\(2^x \leq 4\)

\(2^x \leq 2^2\) → \(x \leq 2\)

Domain: \(x \leq 2\)

Example 9: \(f(x) = \sqrt{e^x - e^{-x}}\)

Condition: \(e^x - e^{-x} \geq 0\)

\(e^x \geq e^{-x}\)

Multiply by\(e^x\) (positive): \(e^{2x} \geq 1\)

\(2x \geq 0\) → \(x \geq 0\)

Domain: \(x \geq 0\)

Example 10: \(f(x) = \frac{1}{2^x - 1}\)

Condition: \(2^x - 1 \neq 0\)

\(2^x \neq 1\) → \(x \neq 0\)

Domain: \(x \neq 0\)

Example 11: \(f(x) = \frac{x}{e^x - e^2}\)

Condition: \(e^x - e^2 \neq 0\)

\(e^x \neq e^2\) → \(x \neq 2\)

Domain: \(x \neq 2\)

Example 12: \(f(x) = \frac{2^x}{3^x - 9}\)

Condition: \(3^x - 9 \neq 0\)

\(3^x \neq 9 = 3^2\) → \(x \neq 2\)

Domain: \(x \neq 2\)

Example 13: \(f(x) = \frac{1}{\sqrt{e^x - 2}}\)

Condition (root in denominator): \(e^x - 2 > 0\)

\(e^x > 2\)

\(x > \ln(2)\)

Domain: \(x > \ln(2)\)

Example 14: \(f(x) = \sqrt{x} \cdot e^x\)

Root condition: \(x \geq 0\)

The exponential: no restriction

Domain: \(x \geq 0\)

Example 15: \(f(x) = \frac{e^x}{\sqrt{x - 1}}\)

Root-in-denominator condition: \(x - 1 > 0\) → \(x > 1\)

Domain: \(x > 1\)