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Mathematics

What is the domain of a function?

The domain of a function is all of the allowed input values of a function. The domain can be defined in a number of different ways. Here are some examples. - Using set notation - $x=\{2,3,4,5\}$, this tells us that $x$, the input to the function, can be any of the numbers in the curly braces. - $x \in \mathbb{R}, x \neq -2 \hspace{1em}$ this means $x$, the input to the function, can be any real number except $-2$. - Using inequalities - $-1 < x \leq 4$ where $x \in \mathbb{R}\hspace{1em}$ this means $x$, the input, can be any real number larger than $-1$ and less than or equal to $4$. - $x > 5$ this means $x$, can be any real number larger than $5$. - Using words - $x$ can be any integer less than or equal to $-2$. Sometimes we need to identify the domain from given information. For example, if we are given the graph of a function, we can identify the domain by looking at the $x$-axis. The end-points of the function can either be a - Filled circle $\rightarrow$ indicating that the end-point is included in the domain - Open circle $\rightarrow$ indicating that the end-point is __not__ included in the domain - Arrow $\rightarrow$ indicating that the domain tends to infinity If we are dealing with a rational (fractional) function then any $x$-values that make the denominator of the function equal zero must be excluded from the domain. For example, if $f(x)=\dfrac{1}{x-3}$ then the domain could be $x \in \mathbb{R}, x \neq 3$, the domain restriction of $x \neq 3$ would be represented by a vertical asymptote with equation $x=3$.

Psychology

What should you do to increase the reliability of an experiment?

Reliability refers to the extent to which results or measurements remain consistent when research is repeated by the original researchers (repeatability) or by different researchers using the same methods (reproducibility). A researcher can do the following to increase the reliability of an experiment. **Standardise procedures**: Using the same instructions, materials, and conditions for all participants reduces variation caused by inconsistent administration. This ensures that any repetition of the experiment is more likely to yield consistent results, directly increasing reliability. **Use reliable measurement tools**: Instruments that consistently produce the same results (e.g. validated tests, calibrated equipment) reduce measurement error. This increases the reliability of data across repeated uses. **Repeat the experiment (supports repeatability)**: Conducting the experiment more than once checks whether the same researchers can get consistent results under the same conditions. This directly tests and improves repeatability, a key aspect of reliability. **Clearly document methods and procedures (supports reproducibility)**: Providing detailed descriptions of how the study was conducted allows other researchers to accurately repeat it. This improves reproducibility by enabling others to get consistent results using the same method.

Environmental Systems and Societies

How does latitude affect climate?

In general, the farther you move from the Equator, the colder the climate becomes. At 0° latitude (the Equator), the climate is warm, while at 90° latitude (the Poles), it is cold. However, climate is actually shaped by insolation (sunlight received), temperature, and precipitation, which show distinct trends as latitude increases. Let’s examine how these three factors vary at different latitudes. At the Equator, the Sun’s rays strike the Earth almost directly throughout the year. As a result, insolation and temperatures remain high all year. The strong solar energy heats the air, increasing evaporation. The resulting water vapour leads to a humid climate with heavy rainfall. Water vapour also acts as a greenhouse gas, trapping heat and further raising temperatures. Countries such as **Ecuador, Brazil, Uganda**, and **Indonesia** all have warm tropical climates with high rainfall. At about 30° latitude, the Sun’s rays hit directly at the height of summer but at an angle of roughly 30° in the middle of winter. This creates hot summers and cooler winters, with insolation and temperature varying by season. Due to the Hadley circulation, air descending at this latitude is dry and causes low rainfall. As a result, these regions are dominated by **deserts**, with low precipitation, dry air, and extreme day–night temperature swings. At around 40° latitude, the Sun’s rays always strike at an angle, with the steepest angle in summer and the shallowest in winter. This creates warm summers and cold winters. Here, the Ferrell cells move heat and moisture poleward, producing moderate to high rainfall. Cities such as **Beijing, Istanbul, Madrid**, and **New York** sit at about 40° North, while **Argentina, Chile, Wellington (New Zealand)**, and **Tasmania (Australia)** lie near 40° South. These areas experience seasonal climates with significant precipitation, but local features such as proximity to oceans or mountains strongly influence the exact climate. At around 66° latitude, we reach the Arctic Circle (North) and the Antarctic Circle (South). From here until the Poles, the tilt of the Earth creates polar days (24 hours of sunlight) at the height of summer and polar nights (24 hours of darkness) at the height of winter. The extremely low insolation results in very cold temperatures. Precipitation is also minimal because evaporation is weak, and most moisture is already released around 60° latitude by the Polar cells, effectively creating polar deserts in the Arctic and Antarctic.

Biology

How to differentiate between population density and population distribution?

Population density asks “How many?” while population distribution answers “Where and how are organisms spread?” Each describes a different pattern in space. Understanding the difference is important when studying ecosystems. Population density refers to the number of individuals of a species per unit area or volume. It is a numerical measure that tells you how crowded an area is. For example, a forest with 100 deer spread across 10 square kilometers has a population density of 10 deer per square kilometer. This measurement is useful for understanding the intensity of competition for resources, such as food, water, or territory, within a given habitat. In contrast, population distribution describes the pattern or spatial arrangement of individuals across a given area. It tells you how the population is spread out, not just how many there are. There are three main types of distribution patterns: uniform, where individuals are evenly spaced (often due to territorial behavior or competition); clumped, where individuals group in patches (often for social reasons or because resources are unevenly distributed); and random, where the position of one individual is independent of another (usually in habitats with abundant resources and little competition). To illustrate the difference, imagine two regions with the same population density of 50 people per square kilometer. In one region, those people may live evenly spaced across the land (uniform distribution), while in another, they may be concentrated in towns with empty countryside in between (clumped distribution). The density is the same, but the distribution reveals much more about how the population interacts with the environment and each other. In short, population density is a quantity—a measure of how many individuals exist in a specific area—while population distribution is a pattern—a description of how those individuals are arranged. Both are essential tools in studying populations, but they answer different questions: "How many?" versus "Where and how are the organisms distributed or spread?"

Mathematics

What is the change of base formula?

The change of base formula, shown below, is used to change the base of a logarithm from one value to another. $$ \begin{align*} \log_a{x}=\frac{\log_b{x}}{\log_b{a}} \end{align*} $$ Where $a,b \neq 1$ and $x>0$. We can transform a logarithm with base $a$ into the expression on the R.H.S. with base $b$. The formula is particularly useful as we can choose any value we like for $b$. We can use the formula for a range of different things, such as - simplifying expressions with logarithms - solving equations with logarithms - comparing the size/value of logarithms with different bases For example, if we want to simplify $\log_2{x}+\log_4{y}$ we need to change one of the terms in the expression so that they both have the same base. This then allows us to use other logarithm laws. Let's use the formula to change the logarithm in the second term to base $2$. So that means we choose $b=2$. Using other well-known logarithm laws, we get. $$ \begin{align*} \log_2{x}+\log_4{y}&=\log_2{x}+\frac{\log_2{y}}{\log_2{4}}\hspace{2em}(\text{recall }\rightarrow 2^2=4)\\[12pt] &=\log_2{x}+\frac{\log_2{y}}{2}\\[12pt] &=\log_2{x}+\log_2{\sqrt{y}}\\[12pt] &=\log_2{x\sqrt{y}} \end{align*} $$

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