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Power Functions

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

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Power Functions

1. Core Concepts

**Power Functions** are functions of the form $y = x^{\alpha}$, where $x$ is the variable and $\alpha$ is a constant.

**Key Distinction**:

* **Power Function**: Base is variable $x$, Exponent is constant (e.g., $y=x^3$).

* **Exponential Function**: Base is constant $a$, Exponent is variable $x$ (e.g., $y=3^x$).

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2. Classification of Graphs and Properties

The shape depends on $\alpha$. In the first quadrant, all power functions pass through $(1, 1)$.

#### (1) $\alpha > 0$ (Passes through Origin)

* **Property**: Passes through $(0,0)$ and $(1,1)$. **Increasing** on $(0, +\infty)$.

* **Shape Details**:

* $\alpha > 1$ (e.g., $x^2, x^3$): **Concave Up**. Grows faster. Below $y=x$ in $(0,1)$, above in $(1, +\infty)$.

* $\alpha = 1$ (e.g., $x$): Straight line.

* $0 < \alpha < 1$ (e.g., $x^{1/2}$): **Concave Down**. Grows slower. Above $y=x$ in $(0,1)$, below in $(1, +\infty)$.

#### (2) $\alpha < 0$ (Does NOT pass through Origin)

* **Property**: Passes through $(1,1)$. **Decreasing** on $(0, +\infty)$. The axes are **asymptotes**.

* **Example**: $y = x^{-1} = \frac{1}{x}$ (Hyperbola).

#### (3) Parity (Even/Odd)

Depends on $\alpha$:

* Even integer $\alpha$ $\rightarrow$ **Even Function** (Symmetric about y-axis).

* Odd integer $\alpha$ $\rightarrow$ **Odd Function** (Symmetric about origin).

* Fractional $\alpha$ $\rightarrow$ Often **Neither** (due to domain constraints).

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3. Relative Position of Graphs

To compare graphs, look at the region where $x > 1$. A larger $\alpha$ means the graph is higher (steeper).

**Rule**: For $x>1$, graphs are arranged from bottom to top as $\alpha$ increases.

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4. Typical Examples

**Example 1: Domain**

Find domain of $f(x) = (x-2)^{\frac{1}{3}} + (3x-1)^{-2}$.

**Solution**:

1. $(x-2)^{\frac{1}{3}}$: Odd root, domain is $\mathbb{R}$.

2. $(3x-1)^{-2}$: Negative exponent implies denominator $\neq 0 \Rightarrow x \neq \frac{1}{3}$.

**Result**: $(-\infty, \frac{1}{3}) \cup (\frac{1}{3}, +\infty)$.

**Example 2: Comparison**

Compare $3.1^{0.4}$ and $3.2^{0.4}$.

**Solution**:

Use function $y = x^{0.4}$. Since $\alpha = 0.4 > 0$, it is increasing.

Since $3.1 < 3.2$, then $3.1^{0.4} < 3.2^{0.4}$.

**Example 3: Graph Identification**

Match curves $C_1, C_2, C_3$ to $y=x^3, y=x, y=x^{\frac{1}{3}}$.

**Solution**:

Look at $x > 1$:

* Highest curve $C_1$ corresponds to largest $\alpha=3$.

* Middle curve $C_2$ corresponds to $\alpha=1$.

* Lowest curve $C_3$ corresponds to smallest $\alpha=\frac{1}{3}$.

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5. Common Pitfalls

* **Confusion**: $y=x^2$ is Power, $y=2^x$ is Exponential. Check where the $x$ is.

* **Domain**: For $y=x^{1/2}$ or $y=x^{-1}$, remember $x$ cannot be negative or zero respectively.

* **Monotonicity Notation**: For $\alpha < 0$, never describe the decreasing interval using union $\cup$. Say "decreasing on $(-\infty, 0)$ AND $(0, +\infty)$ separately".