지수함수와 로그함수 전체 문제

In the following exercises, solve for x, giving an exact answer as well as an approximation to three decimal places.

${\left(\frac{1}{2}\right)}^{x}=10$

In the following exercises, solve for x, giving an exact answer as well as an approximation to three decimal places.

$7{e}^{x-3}=35$

In the following exercises, solve for x, giving an exact answer as well as an approximation to three decimal places.

$8{e}^{x+5}=56$

Sung Lee invests $\char36 5,000$ at age 18. He hopes the investments will be worth $\char36 10,000$ when he turns 25. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Is that a reasonable expectation?

Alice invests $\char36 15,000$ at age 30 from the signing bonus of her new job. She hopes the investments will be worth $\char36 30,000$ when she turns 40. If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal?

Coralee invests $\char36 5,000$ in an account that compounds interest monthly and earns $7\%.$ How long will it take for her money to double?

Simone invests $\char36 8,000$ in an account that compounds interest quarterly and earns $5\%.$ How long will it take for his money to double?

Carbon-14 is used for archeological carbon dating. Its half-life is 5,730 years. How much of a 100-gram sample of Carbon-14 will be left in 1000 years?

What is the difference between the equation for exponential growth versus the equation for exponential decay?

Find and Evaluate Composite Functions

In the following exercises, for each pair of functions, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

$f(x)=7x-2$ and $g(x)=5x+1$

Find and Evaluate Composite Functions

In the following exercises, for each pair of functions, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

$f(x)=4x$ and $g(x)={x}^{2}+3x$

In the following exercises, evaluate the composition.

For functions $f(x)=3{x}^{2}+2$ and $g(x)=4x-3,$ find ⓐ $(f\circ g)(-3)$ ⓑ $(g\circ f)(-2)$ ⓒ $(f\circ f)(-1)$

In the following exercises, evaluate the composition.

For functions $f(x)=2{x}^{3}+5$ and $g(x)=3{x}^{2}-7,$ find ⓐ $(f\circ g)(-1)$ ⓑ $(g\circ f)(-2)$ ⓒ $(g\circ g)(1)$

Determine Whether a Function is One-to-One

In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

$\{(-3,-5),(-2,-4),(-1,-3),(0,-2),$ $(-1,-1),(-2,0),(-3,1)\}$

Determine Whether a Function is One-to-One

In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

$\{(-3,0),(-2,-2),(-1,0),(0,1),$ $(1,2),(2,1),(3,-1)\}$

Determine Whether a Function is One-to-One

In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

$\{(-3,3),(-2,1),(-1,-1),(0,-3),$ $(1,-5),(2,-4),(3,-2)\}$

Find the Inverse of a Function

In the following exercise, find the inverse of the function. Determine the domain and range of the inverse function.

$\{(-3,10),(-2,5),(-1,2),(0,1)\}$

In the following exercises, find the inverse of each function.

$f(x)=6x-11$

In the following exercises, find the inverse of each function.

$f(x)={x}^{3}+13$

In the following exercises, find the inverse of each function.

$f(x)=\frac{1}{x+5}$