The constant second difference of a quadratic number pattern is - 2. The pattern has a maximum value of 44 at the 8th term. Determine the general term of this quadratic pattern in the form Tn = an² + bn + c.

The constant second difference of a quadratic number pattern is -2. The pattern has a maximum value of 44 at the 8th term. Determine the general term of this quadratic pattern in the form $T_n = an^2 + bn + c$.

There was a hole at the bottom of a rusty water tank. The hole became bigger as water passed through it. A certain amount of water passed through the hole at the end of the first minute. At the end of each minute thereafter, 2 more litres of water passed through the hole than at the end of the previous minute.
If the tank was half full at the end of 36 minutes and empty at the end of 54 minutes, how many litres of water passed through the hole at the end of the first minute?

Explain why $f(x)=5^{-x}$ is a decreasing function.

The function $f(x) = 5^{-x}$ can be rewritten as $f(x) = \left(\frac{1}{5}\right)^x$. For an exponential function of the form $g(x) = b^x$, if the base $b$ is between 0 and 1 (i.e., $0 < b < 1$), the function is decreasing. In this case, the base is $\frac{1}{5}$, which satisfies $0 < \frac{1}{5} < 1$. Therefore, as $x$ increases, the value of $f(x)$ decreases.

Determine $f^{-1}(x)$, the inverse of $f(x)$, in the form $y=...$

Draw the graphs of $f$ and $f^{-1}$ on the same set of axes provided in the ANSWER BOOK. Clearly show the asymptotes and intercept(s) with the axes for EACH graph.

Since I cannot draw, I will list the key features for drawing the graphs:

For $f(x) = 5^{-x}$:

• Horizontal Asymptote: $y=0$ (the x-axis). As $x \to \infty$, $5^{-x} \to 0$.
• y-intercept: Set $x=0 \implies f(0) = 5^0 = 1$. So, the y-intercept is $(0,1)$.
• x-intercept: Set $f(x)=0 \implies 5^{-x}=0$, which has no solution. There is no x-intercept.
• Shape: It is a decreasing exponential function.
• Additional points (optional for sketching): $f(-1)=5$, $f(1)=\frac{1}{5}$.

For $f^{-1}(x) = -\log_5 x$:

• Vertical Asymptote: $x=0$ (the y-axis). The domain of $\log_5 x$ is $x>0$. As $x \to 0^+$, $\log_5 x \to -\infty$, so $-\log_5 x \to \infty$.
• x-intercept: Set $f^{-1}(x)=0 \implies -\log_5 x = 0 \implies \log_5 x = 0 \implies x = 5^0 = 1$. So, the x-intercept is $(1,0)$.
• y-intercept: Set $x=0 \implies -\log_5 0$, which is undefined. There is no y-intercept.
• Shape: It is a decreasing logarithmic function.
• Additional points (optional for sketching): $f^{-1}(5) = -\log_5 5 = -1$, $f^{-1}(\frac{1}{5}) = -\log_5 (\frac{1}{5}) = -(-1) = 1$.

The graphs of $f(x)$ and $f^{-1}(x)$ are reflections of each other across the line $y=x$.

Write down the values of $x$ for which $f^{-1}(x) \ge -1$.