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Basic Elementary Functions

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

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Basic Elementary Functions

1. Overview

Basic elementary functions are the fundamental "building blocks" for the CSCA exam. They are rarely tested on definitions alone but are used to test **Domain, Range, Monotonicity, Parity**, and **Graph Transformations**. The four main categories are:

1. Power Functions

2. Exponential Functions

3. Logarithmic Functions

4. Trigonometric Functions

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2. Power Functions

**Formula**: $y = x^{\alpha}$ ($\alpha$ is a constant).

**Exam Focus**: Master the graphs of five common types: $\alpha \in \{1, 2, 3, \frac{1}{2}, -1\}$.

* **Commonality**: All pass through the point $(1, 1)$.

* **Monotonicity**:

* $\alpha > 0$: Increasing on $(0, +\infty)$.

* $\alpha < 0$: Decreasing on $(0, +\infty)$ (Axes are asymptotes).

* **Parity**: Depends on $\alpha$ (e.g., $x^2$ is Even, $x^3$ is Odd).

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3. Exponential & Logarithmic Functions

These are **Inverse Functions** of each other. Their graphs are symmetric about the line $y=x$. The **Base $a$** determines the growth/decay behavior.

| Feature | Exponential $y = a^x$ | Logarithmic $y = \log_a x$ |

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

| **Condition** | $a > 0, a \neq 1$ | $a > 0, a \neq 1$ |

| **Domain** | $\mathbb{R}$ | $(0, +\infty)$ |

| **Range** | $(0, +\infty)$ | $\mathbb{R}$ |

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

| **Trend** | $a>1$ Inc.; $0<a<1$ Dec. | $a>1$ Inc.; $0<a<1$ Dec. |

**Exam Tip**:

* When solving inequalities, check if the base is $>1$. If $0<a<1$, flip the inequality sign.

* Remember the symmetry: Log is the mirror image of Exp across $y=x$.

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4. Trigonometric Functions

**Exam Focus**: Graph shapes, Period, Symmetry, and Domain restrictions.

| Function | $y = \sin x$ | $y = \cos x$ | $y = \tan x$ |

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

| **Period** | $2\pi$ | $2\pi$ | $\pi$ |

| **Parity** | Odd (Origin sym.) | Even (y-axis sym.) | Odd |

| **Range** | $[-1, 1]$ | $[-1, 1]$ | $\mathbb{R}$ |

| **Domain Trap** | $\mathbb{R}$ | $\mathbb{R}$ | $x \neq k\pi + \frac{\pi}{2}$ (Asymptotes) |

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5. Common Pitfalls

1. **Domain Oversight**: For $\log_a f(x)$ (argument $>0$) or $\tan f(x)$, always define the domain first.

2. **Base Cases**: For $y=a^x$, if $a$ is unknown, you MUST discuss cases $a>1$ and $0<a<1$ separately.

3. **Composite Monotonicity**: For $y = \log_a(g(x))$, apply the "Same-Increasing, Different-Decreasing" rule, but restrict it to the valid domain ($g(x)>0$).