Georgia Milestones Algebra I Building Functions. Practice it free.

Set $y=2x^{3}+1$. Swap: $x=2y^{3}+1$. Subtract $1$: $x-1=2y^{3}$. Divide by $2$: $\dfrac{x-1}{2}=y^{3}$. Take the cube root: $y=\sqrt[3]{\dfrac{x-1}{2}}$. So $f^{-1}(x)=\sqrt[3]{\dfrac{x-1}{2}}\textsf{.}$

Adding $7$ outside the function increases every output by $7$, so the graph shifts $7$ units up.

Multiplying the outputs by $2$ doubles every $y$-value and leaves the $x$-values unchanged. The graph is stretched vertically by a factor of $2$.

A downward shift of $6$ replaces $y$ with $y + 6$. At the $y$-intercept, $x = 0$, so $2(y + 6) = 20$. Then $2y + 12 = 20$, $2y = 8$, and $y = 4$.

$3\log a = \log(a^{3})$ is the power rule.

The common difference is $d = 4$. With $a_1 = 5$, the $20$th term is $a_{20} = 5 + 19(4) = 81$.

Work from the outside in. Since $p(4) = 5$, we need $q(r(x)) = 4$. Since $q(3) = 4$, we need $r(x) = 3$. From the table, $r(4) = 3$, so $x = 4$.

Following the pattern:
1st term, or $n = 1\textsf{:}$ $4 \cdot 5^{0}$
2nd term, or $n = 2\textsf{:}$ $4 \cdot 5^{1}$
3rd term, or $n = 3\textsf{:}$ $4 \cdot 5^{2}$
4th term, or $n = 4\textsf{:}$ $4 \cdot 5^{3}$
Since each term has an exponent that is $1$ less than the term number, the $n$th term is $f(n) = 4 \cdot 5^{n-1}$.