The exponential function

Determine the domain, range and equation of the horizontal asymptote of the exponential function $f(x) = 2,5^x$. Does this function cross the x-axis at any point? Give reasons for your answer.

Complete a table of values for the function $f(x) = 5^{-x}$ using integer values of $x$ from $-2$ to $2$. Write an exponential function equivalent to $f(x)$ but with a positive exponent.

Calculate the average rate of change of the exponential function $f(x) = 3^x$ in the intervals $[1, 2]$ and $[2, 3]$. What do you observe when comparing both results? What does this indicate about the rate of growth of the exponential function compared to a linear function?

State which of the following algebraic expressions correspond to an exponential function and justify your answer:

1. a) $f(x) = x^3$
2. b) $g(x) = 3^x$
3. c) $h(x) = 3x$
4. d) $i(x) = \left(\frac{1}{4}\right)^x$
5. e) $j(x) = 1^x$

Starting from the function $f(x) = 2^x$, explain what geometric transformations (translations) must be applied to obtain the graph of the function $g(x) = 2^{x-3} + 4$. State the equation of the horizontal asymptote of the function $g(x)$.

The following graph corresponds to an exponential function of the form $f(x) = 2^{x-p} + q$. By observing the horizontal asymptote and the marked points on the curve, determine the values of the parameters $p$ and $q$, and write the complete algebraic expression of the function.

A family deposits $6000$ euros into a savings account offering an annual compound interest rate of $2,5\%$. Write the function $C(t)$ expressing the accumulated capital after $t$ years. What will the account balance be after $8$ years if no transactions are made? Round the result to two decimal places.

A radioactive substance loses half of its mass every $15$ days. If initially there is a sample of $200$ grams, write the function $M(t)$ that models the remaining mass in terms of time $t$ (in days). How many grams will remain exactly after $60$ days?

The price of a technological component depreciates over time according to an exponential function of the form $V(t) = k \cdot a^t$, where $t$ is measured in years. If the component cost $800$ euros when new ($t=0$) and after $2$ years its value is $450$ euros, calculate the exact values of $k$ and $a$. What will its value be after $4$ years?

Determine and justify for which values of the parameter $m$ the function $f(x) = \left(\frac{2m - 5}{3}\right)^x$ represents a strictly decreasing exponential function. Take into account the algebraic conditions that the base of such an exponential function must satisfy.