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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 $x=3$.

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. $$ \begin{align*} \log_2{x}+\log_4{y}&=\log_2{x}+\frac{\log_2{y}}{\log_2{4}}\hspace{2em}(\text{recall }\rightarrow 2^2=4)\\[12pt] &=\log_2{x}+\frac{\log_2{y}}{2}\\[12pt] &=\log_2{x}+\log_2{\sqrt{y}}\\[12pt] &=\log_2{x\sqrt{y}} \end{align*} $$

Physics

How to find total mechanical energy?

Mechanical energy is defined as the sum of an object’s energies due to its motion and position. The amount of mechanical energy an object possesses determines the amount of work it can do on other objects, or in other words, the amount of energy it is capable of transferring to other objects. Given the above definition, we can be more specific about the types of energy that contribute to the total mechanical energy of an object: - Kinetic energy $E_k$ is the energy associated with an object’s motion and is given by the formula $E_k = \dfrac{1}{2}mv^2$ where $m$ is the mass of the object and $v$ is its speed. Note that this is the macroscopic energy of motion of the particle, and does not include the microscopic energy of the movement of its particles which contributes to the object's internal energy. The energy associated with an object’s position is called its potential energy. Two main types of potential energy will contribute to mechanical energy: - Gravitational potential energy $E_p$ is the energy of an object due to its position in a gravitational field. For an object at a height $h$ above a reference point, its gravitational potential energy is calculated with the formula $E_p = mgh$, where $m$ is the mass of the object and $g$ is the gravitational field strength. At the Earth’s surface, the value for $g$ is 9.8 N kg$^{-1}$. - Elastic potential energy $E_H$ is is the energy stored due to the deformation of an elastic object. Work can be done in changing the shape of an object, for example stretching or compressing a string, and energy is stored in the object as a result. This stored energy is released when the object returns to its original shape. The standard formula for elastic potential energy is derived from Hooke's Law: $E_H = \dfrac{1}{2}kx^2$. Where $k$ is the spring constant and $x$ is the displacement from the equilibrium position. Another potential energy that can be considered to contribute to total mechanical energy is electric potential energy. Like gravitational potential energy, a charged object will have stored energy due to its position in an electric field. Because the idea of mechanical energy is normally applied to larger objects and not small charged particles, we will ignore it here. Having discussed the different types of energy that contribute to the mechanical energy of an object, we can create a formula for total mechanical energy. In words, the formula is $\hspace{3em} $ Mechanical Energy = Kinetic Energy + Potential Energy or more specifically $\hspace{3em} $ Mechanical Energy = Kinetic Energy + Gravitational Potential energy + Elastic Potential Energy In the form of an equation, the total mechanical energy can be expressed as $\hspace{3em} E_{tot}=E_k + E_p + E_H$ or $\hspace{3em} E_{tot}=\dfrac{1}{2}mv^2+ mgh+ \dfrac{1}{2}kx^2$

Economics

How does an increase in net exports shift the AD curve?

Aggregate demand (AD) is the total demand for an economy’s output at a given price level. It has four components: - Consumption (C): household spending on goods and services. - Investment (I): business spending on capital goods and raw materials. - Government spending (G): government purchases of goods and services (not transfers). - Net exports (NX): the value of exports (X) minus the value of imports (M). When any of these components increase, ceteris paribus, AD increases, shown by a rightward shift of the AD curve. For C, I, G, and exports, the relationship is straightforward: more spending on domestic output means higher AD. But it is less obvious why a fall in imports raises AD. Consider this example: If German households buy 5 German cars and 1 Japanese car, German output is 5 cars. Yet firms report sales as 6 cars, so consumption is recorded as the value of 6 cars. To avoid overstating domestic output, the value of the imported car is subtracted. In effect, imports are deducted from C, I, and G to ensure AD reflects demand for domestic production only. This is especially important with imported inputs. If a $\text{\textdollar}$100 belt is recorded as consumption but $\text{\textdollar}$20 of materials were imported, subtracting imports ensures AD reflects the $\text{\textdollar}$80 of domestic value added. Returning to the car example: if next year total car sales stay at 6 but imports fall to zero, consumption is unchanged, but domestic production must have risen from 5 to 6. In this case, households are substituting a domestically produced car for the previously imported one. Thus, lower imports—holding other components constant—mean greater demand for domestic output. In short: - If exports increase, foreign demand for domestic goods rises → AD increases. - If imports decrease (without a fall in C, I, or G), households or firms must be substituting toward domestic goods → AD increases. Therefore, when net exports increase—whether from higher exports or lower imports—aggregate demand increases, and the AD curve shifts to the right.

Psychology

How does culture influence your goals?

Culture can influence a person's goals through the cultural dimension of individualism vs. collectivism, which refers to how much a culture values personal independence (individualism) versus group cohesion and interdependence (collectivism). This dimension exists on a continuum, meaning some cultures are high in individualism or collectivism, while others may be more moderate or low. In more individualistic cultures (e.g., the USA), people are encouraged to pursue personal goals, such as career success, self-improvement, or personal happiness. Individuals in these cultures often define their identity based on personal traits and achievements, and success is typically seen as the result of one’s own efforts. In contrast, in more collectivistic cultures (e.g., China or Japan), individuals are more likely to set goals that reflect the needs and expectations of their family, community, or social group. People may prioritise group harmony, social roles, and responsibilities over personal ambition. For example, a person might choose a career path that benefits their family or upholds group traditions, even if it doesn’t align with their personal preference.

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