What are the definitions and properties of the following: identity function, explicit function, hyperbolic identities, hyperbolic cosecant, limits involving powers, the constant e, and function notati

Here are the solutions to the multiple-choice questions:

1. The function $f(x) = x$ is a linear function (its graph is a straight line), an identity function (it maps every element to itself), and an odd function ($f(-x) = -x = -f(x)$). Therefore, all the given options are correct.

• D) All are correct

2. If $y$ is expressed directly in terms of $x$ as $y = f(x)$, it means $y$ is explicitly defined by $x$. This is called an explicit function.

• A) An explicit function

3. The fundamental hyperbolic identity is analogous to the Pythagorean identity for trigonometric functions. It states that $\cosh^2 x - \sinh^2 x = 1$.

• C) 1

4. The hyperbolic cosecant function, $\text{cosech } x$, is the reciprocal of the hyperbolic sine function, $\sinh x$.
We know that $\sinh x = \frac{e^x - e^{-x}}{2}$.
Therefore,

$$\text{cosech } x = \frac{1}{\sinh x} = \frac{1}{\frac{e^x - e^{-x}}{2}} = \frac{2}{e^x - e^{-x}}$$

• C) $\frac{2}{e^x - e^{-x}}$

5. This is a standard limit form $\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}$.
For the given limit, $n=3$.

$$\lim_{x \to a} \frac{x^3 - a^3}{x - a} = 3a^{3-1} = 3a^2$$

• B) $3a^2$

6. This is the definition of the mathematical constant $e$.

$$\lim_{x \to 0} (1 + x)^{1/x} = e$$

• B) E

7. The notation $y = f(x)$ was introduced by Leonhard Euler in the 18th century.

• B) Euler

8. To find $f(0)$, substitute $x=0$ into the function $f(x) = x^2 - 2x + 1$.

$$f(0) = (0)^2 - 2(0) + 1 = 0 - 0 + 1 = 1$$

• C) 1

9. When we define a function $f$ from set $X$ to set $Y$, the set $X$ containing all possible input values is called the domain of $f$.

• A) Domain of f

10. The identity function is $f(x) = x$. Its graph is a straight line that passes through the origin $(0,0)$ and makes a $45^\circ$ angle with the positive x-axis, thereby bisecting the first and third quadrants.

• A) Bisecting the $1^\text{st}$ and $3^\text{rd}$ quad

11. If $f(x) = x^2$, the output values (range) are always non-negative, because the square of any real number is greater than or equal to zero.
The range is $[0, \infty[$.

• A) $[0, \infty[$

12. For the function $f(x) = \frac{x}{x^2 - 4}$, the denominator cannot be zero.

$$x^2 - 4 \ne 0$$ $$x^2 \ne 4$$ $$x \ne \pm 2$$

Therefore, the domain consists of all real numbers except $-2$ and 2.

• C) All real numbers except -2, 2

13. For a graph to represent a function, it must pass the vertical line test. This means that any vertical line drawn through the graph must intersect it at exactly one point. If it intersects at more than one point, it means there are multiple $y$-values for a single $x$-value, which is not allowed for a function.

• A) One point only

14. The function is defined in two parts:

1. $f(x) = x$ for $0 \le x \le 1$. This interval is $[0, 1]$.
2. $f(x) = x-1$ for $1 < x < 2$. This interval is $(1, 2)$. The domain of $f$ is the union of these two intervals: $[0, 1] \cup (1, 2) = [0, 2)$. Among the given options, $(0,2)$ is the closest, although it incorrectly excludes $x=0$. Given the options, it is the most likely intended answer.

• A) $(0,2)$

15. A linear function is defined by an equation of the form $y = mx + c$, where $m$ and $c$ are constants. The graph of any such equation is always a straight line.

• A) Straight line

16. The identity function $f: X \to X$ maps every element of set $X$ to itself. Therefore, its domain is $X$ (the set of all possible inputs) and its range is also $X$ (the set of all possible outputs).

• A) X

17. A linear function $f(x) = ax + b$ is a constant function if its value does not change with $x$. This occurs when the coefficient of $x$ is zero.
If $a=0$, then $f(x) = 0 \cdot x + b = b$, which is a constant value.

• D) $a=0$