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Exponential Functions

CSCA Exponential Functions study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Exponential Functions

1. Core Concepts

**Definition**: A function $y = a^x$, where $x$ is the variable, base $a$ is a constant, $a > 0, a \neq 1$.

**Why limit the base?**

* If $a < 0$, values like $a^{1/2}$ are undefined in real numbers.

* If $a = 1$, $y=1$ is just a constant line.

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2. Graphs and Properties

Properties depend entirely on the range of base $a$: "Growth" ($a>1$) vs. "Decay" ($0<a<1$).

| Feature | **$a > 1$ (e.g., $y=2^x$)** | **$0 < a < 1$ (e.g., $y=0.5^x$)** |

| :--- | :--- | :--- |

| **Shape** | Rising Curve | Falling Curve |

| **Monotonicity** | **Increasing** on $\mathbb{R}$ | **Decreasing** on $\mathbb{R}$ |

| **Fixed Point** | Passes $(0, 1)$ | Passes $(0, 1)$ |

| **Range** | $(0, +\infty)$ | $(0, +\infty)$ |

| **Asymptote** | x-axis ($y=0$) | x-axis ($y=0$) |

| **Trend** | $x \to +\infty, y \to +\infty$ | $x \to +\infty, y \to 0$ |

**Symmetry Rule**:

The graphs of $y=a^x$ and $y=(\frac{1}{a})^x$ are symmetric about the **y-axis**.

(e.g., $y=2^x$ and $y=0.5^x$ are mirror images).

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3. Key Operational Rules

Assume $a, b > 0$ and $r, s \in \mathbb{R}$:

1. $a^r \cdot a^s = a^{r+s}$

2. $\frac{a^r}{a^s} = a^{r-s}$

3. $(a^r)^s = a^{rs}$

4. $(ab)^r = a^r b^r$

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4. Typical Examples

**Example 1: Monotonicity Comparison**

Compare (1) $1.7^{2.5}$ vs $1.7^3$; (2) $0.8^{-0.2}$ vs $0.8^{-0.1}$.

**Solution**:

(1) Base $1.7 > 1$ (Increasing). Since $2.5 < 3$, then $1.7^{2.5} < 1.7^3$.

(2) Base $0.8 < 1$ (Decreasing). Since $-0.2 < -0.1$, the inequality flips: $0.8^{-0.2} > 0.8^{-0.1}$.

**Example 2: Substitution Method**

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

**Solution**:

Note that $4^x = (2^x)^2$. Let $t = 2^x$ (**Condition: $t > 0$**).

Equation becomes $t^2 - 3t + 2 = 0 \Rightarrow (t-1)(t-2)=0$.

* $t=1 \Rightarrow 2^x = 1 \Rightarrow x=0$.

* $t=2 \Rightarrow 2^x = 2 \Rightarrow x=1$.

**Answer**: $x=0, x=1$.

**Example 3: Domain and Inequality**

Find domain of $y = \sqrt{1 - (\frac{1}{2})^{x-1}}$.

**Solution**:

Radicand $\ge 0$: $1 - (\frac{1}{2})^{x-1} \ge 0 \Rightarrow (\frac{1}{2})^{x-1} \le 1$.

Rewrite 1 as $(\frac{1}{2})^0$: $(\frac{1}{2})^{x-1} \le (\frac{1}{2})^0$.

**Crucial Step**: Base $0.5 < 1$, so flip the inequality sign.

$\therefore x - 1 \ge 0 \Rightarrow x \ge 1$.

**Answer**: $[1, +\infty)$.

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5. Common Pitfalls

* **Forgetting to flip sign**: When solving inequalities with base $0 < a < 1$, the direction must reverse.

* **Substitution Range**: When setting $t = a^x$, always remember $t$ must be positive.

* **Graph Confusion**: Don't confuse $y=2^x$ (Exponential) with $y=x^2$ (Power).