指数函数zhǐshù hánshù

Core Concept

An exponential function is a function of the form:

$f(x) = a^x \quad (a > 0, a \neq 1)$

where:

• $a$ is the base (must be positive and not equal to 1)
• $x$ is the exponent (variable)

Key distinction from power function: In exponential functions, the variable is in the exponent; in power functions $x^n$, the variable is in the base.

Domain and Range

• Domain: $\mathbb{R}$ (all real numbers)
• Range: $(0, +\infty)$ (all positive real numbers)

Note: $a^x > 0$ for all real $x$ when $a > 0$.

Key Properties

1. Passes Through (0, 1)

$f(0) = a^0 = 1 \quad \text{for all } a > 0$

Every exponential function passes through the point $(0, 1)$.

2. Always Positive

$a^x > 0 \quad \text{for all } x \in \mathbb{R}$

3. Monotonicity

• If $a > 1$: $f(x) = a^x$ is strictly increasing
• If $0 < a < 1$: $f(x) = a^x$ is strictly decreasing

4. Laws of Exponents

$a^{x+y} = a^x \cdot a^y$ $a^{x-y} = \dfrac{a^x}{a^y}$ $(a^x)^y = a^{xy}$ $(ab)^x = a^x \cdot b^x$

Graph Characteristics

When $a > 1$ (e.g., $y = 2^x$)

• Increases from left to right
• Approaches 0 as $x \to -\infty$
• Grows without bound as $x \to +\infty$
• Horizontal asymptote: $y = 0$

When $0 < a < 1$ (e.g., $y = (1/2)^x$)

• Decreases from left to right
• Grows without bound as $x \to -\infty$
• Approaches 0 as $x \to +\infty$
• Horizontal asymptote: $y = 0$

Comparison of Values

For $a > 1$:

• $a^{x_1} > a^{x_2} \Leftrightarrow x_1 > x_2$

For $0 < a < 1$:

• $a^{x_1} > a^{x_2} \Leftrightarrow x_1 < x_2$

Memory aid: "Base > 1: bigger exponent, bigger value; Base < 1: bigger exponent, smaller value"

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus.

Example 1: Basic (Difficulty ★★☆☆☆)

Compare the values: $2^{0.5}$, $2^{0.3}$, $2^{-0.1}$.

Solution: Since base $2 > 1$, $y = 2^x$ is increasing.

Since $0.5 > 0.3 > -0.1$: $2^{0.5} > 2^{0.3} > 2^{-0.1}$

Answer: $2^{0.5} > 2^{0.3} > 2^{-0.1}$

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Example 2: Intermediate (Difficulty ★★★☆☆)

Compare: $0.5^{-0.1}$, $0.5^{0.1}$, $1.5^{0.1}$.

Solution:

For $0.5^{-0.1}$ and $0.5^{0.1}$: Since $0 < 0.5 < 1$, $y = 0.5^x$ is decreasing. Thus $0.5^{-0.1} > 0.5^{0.1}$.

For $0.5^{0.1}$: This equals $(1/2)^{0.1} = \dfrac{1}{2^{0.1}} < 1$.

For $1.5^{0.1}$: Since $1.5 > 1$ and $0.1 > 0$, we have $1.5^{0.1} > 1^{0.1} = 1$.

For $0.5^{-0.1}$: This equals $2^{0.1} > 1$.

Comparing $2^{0.1}$ and $1.5^{0.1}$: Since $2 > 1.5$ and the exponent is positive: $2^{0.1} > 1.5^{0.1}$

Answer: $0.5^{-0.1} > 1.5^{0.1} > 0.5^{0.1}$

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Example 3: Advanced (Difficulty ★★★★☆)

Find the range of $f(x) = 4^x - 2^{x+1} + 2$, $x \in [-1, 2]$.

Solution:

Let $t = 2^x$. Since $x \in [-1, 2]$: $t \in [2^{-1}, 2^2] = [\dfrac{1}{2}, 4]$

Now $4^x = (2^2)^x = (2^x)^2 = t^2$ and $2^{x+1} = 2 \cdot 2^x = 2t$.

So: $f = t^2 - 2t + 2 = (t-1)^2 + 1$

For $t \in [\dfrac{1}{2}, 4]$:

• Minimum at $t = 1$: $f = 0 + 1 = 1$
• Check endpoints:
  • At $t = \dfrac{1}{2}$: $f = \dfrac{1}{4} - 1 + 2 = \dfrac{5}{4}$
  • At $t = 4$: $f = 16 - 8 + 2 = 10$

Range: $[1, 10]$

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Example 4: Advanced (Difficulty ★★★★☆)

Solve the equation: $4^x - 3 \cdot 2^x + 2 = 0$.

Solution:

Let $t = 2^x$ where $t > 0$.

Then $4^x = (2^x)^2 = t^2$.

Equation becomes: $t^2 - 3t + 2 = 0$

Factor: $(t-1)(t-2) = 0$

So $t = 1$ or $t = 2$.

Back-substitute:

• $2^x = 1 \Rightarrow x = 0$
• $2^x = 2 \Rightarrow x = 1$

Answer: $x = 0$ or $x = 1$

Special Exponential Functions

Natural Exponential Function

$f(x) = e^x \quad \text{where } e \approx 2.71828$

This is the most important exponential function in calculus because $(e^x)' = e^x$.

Common Mistakes

❌ Mistake 1: Confusing with Power Functions

Wrong: $x^2$ is an exponential function ✗

Correct: $2^x$ is exponential (variable in exponent), $x^2$ is a power function ✓

❌ Mistake 2: Wrong Inequality Direction

Wrong: Since $0.5 < 1$, we have $0.5^2 > 0.5^3$ ✗

Correct: For $0 < a < 1$, larger exponent gives smaller value: $0.5^2 < 0.5^3$ ✗

Wait, let me recalculate: $0.5^2 = 0.25$ and $0.5^3 = 0.125$, so $0.5^2 > 0.5^3$ ✓

❌ Mistake 3: Forgetting $a^x > 0$

Wrong: Equation $2^x = -1$ has solution $x = ?$ ✗

Correct: $2^x > 0$ for all $x$, so no solution exists. ✓

Study Tips

1. ✅ Know the two cases: $a > 1$ (increasing) vs $0 < a < 1$ (decreasing)
2. ✅ Use substitution: Let $t = a^x$ to convert to polynomial
3. ✅ Remember the asymptote: $y = 0$ is always the horizontal asymptote
4. ✅ Check: variable in exponent: This distinguishes from power functions

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💡 Exam Tip: When solving exponential equations, use substitution $t = a^x$ to convert to algebraic equations. Remember $t > 0$!