Exponential Functions: Additional Practice (Set 1)

Write $81$ as a power of $3$ and match the exponents.

$3^x = 81 = 3^4 \;\Rightarrow\; x = 4$

Subtract the exponents.

$\frac{6^{x+2}}{6^{x-1}} = 6^{(x+2)-(x-1)} = 6^3 = 216$

Reflecting across the $y$-axis replaces $x$ with $-x$.

$3^{-x} = \left(\tfrac{1}{3}\right)^x$

So the reflected graph is $y = \left(\tfrac{1}{3}\right)^x$, a decay curve.

Take the fourth root, then cube.

$16^{3/4} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$

Any exponential with base greater than $1$ overtakes any power function for large enough $x$, even a slow-looking base like $1.5$.

$1.5^x \;\text{overtakes}\; x^{10} \;\text{eventually}$

Halve until $10$ mg remains.

$80 \to 40 \to 20 \to 10$

That is $3$ halvings, so $3$ years.

Write $9 = 3^2$, so $9^x = 3^{2x}$, and match with $3 = 3^1$.

$3^{2x} = 3^1 \;\Rightarrow\; 2x = 1 \;\Rightarrow\; x = \tfrac{1}{2}$

Increasing $x$ by $4$ multiplies by $2^4$, regardless of the leading $5$.

$\frac{f(x+4)}{f(x)} = 2^4 = 16$

You may raise a positive base to any real power, and the result is always positive.

$\text{domain: all real } x, \qquad \text{range: } y > 0$

The value $f(0) = 6$ gives $a = 6$. Then use $f(1) = 18$.

$6b = 18 \;\Rightarrow\; b = 3$

Subtract the exponents.

$\frac{10^x}{10^{x-2}} = 10^{x-(x-2)} = 10^2 = 100$

Repeatedly multiplying by a base under $1$ shrinks the positive output toward zero.

$b^x \to 0^{+} \;\text{as}\; x \to +\infty$

The curve approaches the asymptote $y = 0$ from above and never goes negative.