Exponential functions

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Exercises

Exercise 1

Given are the functions $f (x) = 2^{x - 2} - 3$ and $g (x) = 4 \cdot {0.5}^{x - 3} - 1$ .

Rewrite both function rules into the form $y = b \cdot g^{t} + d$ . Which transformations do you need to apply to the graphs of the corresponding basic functions to obtain the graphs of $f$ and $g$ ?

Solve algebraically: $f (x) = - 2 \frac{7}{8}$

Solve: $g (x) > 1.5$ . Round the answer to two decimals.

What values can $g (x)$ have for $x \leq 4$ ?

The line $y = p$ intersects the graph of $f$ , but not the graph of $g$ . Compute $p$ .

The line $x = -1$ intersects the graph of $f$ in the point $A$ and the graph of $g$ in the point $B$ . Compute the exact length of the line segment $A B$ .

The line $y = 5$ intersects the graph of $f$ in the point $C$ and the graph of $g$ in the point $D$ . Compute the length of the line segment $C D$ and round the answer to three decimals.

Exercise 2

Solve the following equations and inequalities. Simplify the answer as much as possible and then round the answer to two decimals.

$5^{x} = 10$

$5^{x} \leq 10$

$3 \cdot 5^{x} + 5 = 10$

$3 \cdot 5^{x} + 5 > 10$

${(\frac{1}{3})}^{x} = 2$

${(\frac{1}{3})}^{x} > 2$

$5 \cdot {(\frac{1}{3})}^{x} - 8 = 2$

$5 \cdot {(\frac{1}{3})}^{x} - 8 < 2$

Exercise 3

Solve algebraically:

$2^{x} = 2 \sqrt{2}$

$4^{x} = 8^{x + 2}$

$9^{2 x} = \sqrt{3}$

$2^{2 x - 1} = 32$

$2^{\frac{1}{2} x + 1} = 4 \sqrt{2}$

$2^{2 x} = 1$

$8^{x^{2}} = 4^{2 x}$

${0.5}^{x} = 8$

$3 \cdot 2^{x + 5} = 12$

$32^{3 x - 2} = \frac{1}{4}$

Exercise 4

Solve the following equations and inequalities algebraically:

$5 \cdot 10^{x} = 5000$

$3 \cdot 2^{p} - 2 = 46$

$6 \cdot (5^{t} + 5) = 180$

$162 \cdot {(\frac{1}{3})}^{x} > 2$

$7 + 16 \cdot {1.5}^{x} \leq 43$

$10 \cdot {(\frac{1}{2})}^{x} \geq 160$

Exercise 5

If possible, solve algebraically. Otherwise, give an approximation in two decimals.

$4 \cdot {0.5}^{x} - 1 < 0$

$2 \cdot 2^{- x + 1} - 1 > 0$

$6 \cdot {0.25}^{x} - 4 \geq 0.75$

$3 \cdot {0.5}^{2 x - 1} - 4 < -3.25$

${3.5}^{x + 50} - 0.5 > 3$

$- 2^{x} + 1 \geq -7$

$3^{x - 4} < \frac{1}{9} \sqrt{3}$

Exercise 6

A patient is given medication through a drip. The formula
$A (t) = 540 - 540 \cdot {0.95}^{t}$
gives the amount of the medication $A (t)$ in mg in the patients blood after $t$ minutes.

How can you tell from the formula that the graph of $A (t)$ is increasing?

What is the equation of the asymptote of the graph of $A (t)$ ?

Explain that $A (t)$ does not grow exponentially.

After how many (whole) minutes has $75$ % of the maximum amount of medication entered the blood?

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