Exponential and Logarithmic Functions

Solve the logarithmic equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility. 

$2{\log }_{5}x=3{\log }_{5}4$

The hotel Bora-Bora is having a pig roast. At noon, the chef put the pig in a large earthen oven. The pig’s original temperature was 75°F. At 2:00 pm the chef checked  the pig’s temperature and was upset because it had reached only 100°F. If the oven’s temperature remains a constant 325°F, at what time may the hotel serve its guests, assuming that pork is done when it reaches 175°F?

Begin with the graph of $y=e^x$ and use transformation to graph the function. Determine the domain, range and horizontal asymptote of the function.

$f\left(x\right)=e^{-x}$

Shortest Distance from Chicago to Honolulu Find the shortest distance from Chicago, latitude 41°50 N, longitude

87°37W to Honolulu, latitude 21°18N, longitude 157°50W. Round your answer to the nearest mile.

In Problems 113–116, graph each function. Based on the graph, state the domain and the range and find any intercepts. 

$f\left(x\right)=\left\{\begin{array}{ll}-\ln \left(x\right)& if x<0\\ \ln \left(x\right)& if x>0.\end{array}\right.$

Find $f\left(3x\right)$ if $f\left(x\right)=4-2x^2$.

Find $f\left(3\right)$ if $f\left(x\right)=-4x^2+5x$.

Find $f\left(3x\right)$ if $f\left(x\right)=4-2x^2$.

Find the domain of the function $f\left(x\right)=\frac{x^2-1}{x^2-25}$.

Given two functions f and g, the _____, denoted $f\circ g$ , is defined by $\left(f\circ g\right)\left(x\right)$____.

True or False $f\left(g\left(x\right)\right)=f\left(x\right).g\left(x\right)$.

True or False. The domain of the composite function $\left(f\circ g\right)\left(x\right)$ is the same as the domain of $g\left(x\right)$.

Evaluate each expression using the values given in the table.  

$$\begin{gathered} \left(a\right) \left(f\circ g\right)\left(1\right) \\ \left(b\right) \left(f\circ g\right)\left(-1\right) \\ \left(c\right) \left(g\circ f\right)\left(-1\right) \\ \left(d\right) \left(g\circ f\right)\left(0\right) \\ \left(e\right) \left(g\circ g\right)\left(-2\right) \\ \left(f\right) \left(f\circ f\right)\left(-1\right) \end{gathered}$$

Evaluate each expression using the values given in the table. 

(a) $\left(f\circ g\right)\left(1\right)$ (b) $\left(f\circ g\right)\left(2\right)$ (c) $\left(g\circ f\right)\left(2\right)$ (d) $\left(g\circ f\right)\left(3\right)$ (e) $\left(g\circ g\right)\left(1\right)$ (f) $\left(f\circ f\right)\left(3\right)$.

Evaluate each expression using the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ shown in the figure.

(a) $\left(g\circ f\right)\left(-1\right)$ (b) $\left(g\circ f\right)\left(0\right)$ (c) $\left(f\circ g\right)\left(-1\right)$ (d) $\left(f\circ g\right)\left(4\right)$.

Evaluate each expression using the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ as shown in the given figure.

(a) $\left(g\circ f\right)\left(1\right)$ (b) $\left(g\circ f\right)\left(5\right)$ (c) $\left(f\circ g\right)\left(0\right)$ (d) $\left(f\circ g\right)\left(2\right)$.

Find the value for the given functions $f,g$.

$f\left(x\right)=2x,g\left(x\right)=3x^2+1$

(a) $\left(f\circ g\right)\left(4\right)$ (b) $\left(g\circ f\right)\left(2\right)$ (c) $\left(f\circ f\right)\left(1\right)$ (d) $\left(g\circ g\right)\left(0\right)$.

Find for the given functions$f,g$:

$f\left(x\right)=3x+2,g\left(x\right)=2x^2-1$.

Find : (a) $\left(f\circ g\right)\left(4\right)$ (b) $\left(g\circ f\right)\left(2\right)$ (c) $\left(f\circ f\right)\left(1\right)$ (d) $\left(g\circ g\right)\left(0\right)$.

Find for the given functions$f,g$ 

$f\left(x\right)=4x^2-3,g\left(x\right)=3-\frac{1}{2}{x}^2$.

Find (a) $\left(f\circ g\right)\left(4\right)$ (b) $\left(g\circ f\right)\left(2\right)$ (c) $\left(f\circ f\right)\left(1\right)$ (d) $\left(g\circ g\right)\left(0\right)$.

Find for the given functions $f,g$ :

$f\left(x\right)=2x^2,g\left(x\right)=1-3x^2$.

Find (a) $\left(f\circ g\right)\left(4\right)$ (b) $\left(g\circ f\right)\left(2\right)$ (c) $\left(f\circ f\right)\left(1\right)$ (d) $\left(g\circ g\right)\left(0\right)$.

Find for the given functions $f,g$ such that

$f\left(x\right)=\sqrt{x},g\left(x\right)=2x$.

(a) $\left(f\circ g\right)\left(4\right)$ (b) $\left(g\circ f\right)\left(2\right)$ (c) $\left(f\circ f\right)\left(1\right)$ (d) $\left(g\circ g\right)\left(0\right)$.

Find for the given functions$f,g$ such that

$f\left(x\right)=\sqrt{x+1},g\left(x\right)=3x$.

(a) $\left(f\circ g\right)\left(4\right)$ (b) $\left(g\circ f\right)\left(2\right)$ (c) $\left(f\circ f\right)\left(1\right)$ (d) $\left(g\circ g\right)\left(0\right)$.

Find for the given functions $f,g$ such that

$f\left(x\right)=\left|x\right| , g\left(x\right)=\frac{1}{x^2+1}$.

Find : (a) $\left(f\circ g\right)\left(4\right)$ (b) $\left(g\circ f\right)\left(2\right)$ (c) $\left(f\circ f\right)\left(1\right)$ (d) $\left(g\circ g\right)\left(0\right)$.

Find for the given functions$f,g$ such that

$f\left(x\right)=\left|x-2\right|, g\left(x\right)=\frac{3}{x^2+2}$.

Find : (a)  $\left(f\circ g\right)\left(4\right)$ (b) $\left(g\circ f\right)\left(2\right)$ (c) $\left(f\circ f\right)\left(1\right)$ (d) $\left(g\circ g\right)\left(0\right)$.

Find for the given functions $f,g$ such that

$f\left(x\right)=\frac{3}{x+1} , g\left(x\right)=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$.

Find : (a) $\left(f\circ g\right)\left(4\right)$ (b) $\left(g\circ f\right)\left(2\right)$ (c) $\left(f\circ f\right)\left(1\right)$ (d) $\left(g\circ g\right)\left(0\right)$.

Find for the given functions $f,g$ such that

$f\left(x\right)=x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} , g\left(x\right)=\frac{2}{x+1}$.

Find : (a) $\left(f\circ g\right)\left(4\right)$ (b) $\left(g\circ f\right)\left(2\right)$ (c) $\left(f\circ f\right)\left(1\right)$ (d) $\left(g\circ g\right)\left(0\right)$.

Find the domain of the composite function $(f\circ g)$.

$f\left(x\right)=\frac{3}{x-1} and g\left(x\right)=\frac{2}{x}$

Find the domain of the composite function $f\circ g$.

$f\left(x\right)=\frac{1}{x+3} and g\left(x\right)=-\frac{2}{x}$.

Find the domain of the composite function $f\circ g$.

$f\left(x\right)=\frac{x}{x-1} and g\left(x\right)=-\frac{4}{x}$

Find the domain of the composite function $f\circ g$.

$f\left(x\right)=\frac{x}{x+3} and g\left(x\right)=\frac{2}{x}$

Find the domain of the composite function  $f\circ g$.

$f\left(x\right)=\sqrt{x} and g\left(x\right)=2x+3$

Find the domain of the composite function $f\circ g$.

$f\left(x\right)=x-2 and g\left(x\right)=\sqrt{1-x}$

Find the domain of the composite function $f\circ g$.

$f\left(x\right)=x^2+1 and g\left(x\right)=\sqrt{x-1}$

Find the domain of the composite function $f\circ g$.

$f\left(x\right)=x^2+4 and g\left(x\right)=\sqrt{x-2}$

For the given functions f and g, find: 

$a) f\circ g b) g\circ f c) f \circ f d) g\circ g$

State the domain of each composite function.

$f\left(x\right)=2x+3 and g\left(x\right)=3x$

For the given functions f and g, find:  

$a) f\circ g b) g\circ f c) f \circ f d) g\circ g$

State the domain of each composite function. 

$f\left(x\right)=-x\hspace{0.17em}and\hspace{0.17em}g\left(x\right)=2x-4$

For the given functions f and g, find:   

$a) f\hspace{0.17em}\circ \hspace{0.17em}\hspace{0.17em}g\hspace{0.17em} b)\hspace{0.17em}\hspace{0.17em}g\hspace{0.17em}\circ \hspace{0.17em}\hspace{0.17em}g c) f\hspace{0.17em}\circ \hspace{0.17em}f\hspace{0.17em} d) g\hspace{0.17em}\circ \hspace{0.17em}\hspace{0.17em}f$

State the domain of each composite function. 

$f\left(x\right)=3x+1,\hspace{0.17em}g\left(x\right)=x^2$

For the given functions f and g, find:   

$a) f\circ g b) g\circ f c) f \circ f d) g\circ g$

State the domain of each composite function. 

For the given functions f and g, find:  

$a) f\circ g b) g\circ f c) f \circ f d) g\circ g$

State the domain of each composite function. 

$f\left(x\right)=x^2,\hspace{0.17em}g\left(x\right)=x^2+4$

For the given functions f and g, find:  

$a) f\circ g b) g\circ f c) f \circ f d) g\circ g$

State the domain of each composite function. 

$f\left(x\right)=x^2+1,\hspace{0.17em}g\left(x\right)=2x^2+3$

For the given functions f and g, find:

$a) f\circ g b) g\circ f c) f \circ f d) g\circ g$

State the domain of each composite function. 

$f\left(x\right)=\frac{3}{x-1},\hspace{0.17em}g\left(x\right)=\frac{2}{x}$

In Problems 29 – 44, for the given functions f and g, find:

(a) $f\circ g$

 (b) $g\circ f$

(c) $f\circ f$

 (d)  $g\circ g$

State the domain of each composite function. 

$$\begin{gathered} f\left(x\right)=\frac{1}{x+3}, \\ g\left(x\right)=-\frac{2}{x} \end{gathered}$$

In Problems 29 – 44, for the given functions f and g, find:

(a) $f o g$

 (b) $g o f$

(c) $f o f$

 (d)  $g o g$

$$\begin{gathered} f\left(x\right)=\frac{x}{x-1} \\ g\left(x\right)=-\frac{4}{x} \end{gathered}$$

In Problems 29 – 44, for the given functions f and g, find:

(a) $f\circ g$

 (b) $g\circ f$

(c) $f\circ f$

 (d)  $g\circ g$

State the domain of each composite function.   

$$\begin{gathered} f\left(x\right)=\frac{x}{x+3} \\ g\left(x\right)=\frac{2}{x} \end{gathered}$$

In Problems 29 – 44, for the given functions f and g, find:

(a) $f\circ g$

 (b) $g\circ f$

(c) $f\circ f$

 (d)  $g\circ g$

State the domain of each composite function.   

$$\begin{gathered} f\left(x\right)=\sqrt{x} \\ g\left(x\right)=2x+3 \end{gathered}$$

In Problems 29 – 44, for the given functions f and g, find:

(a) $f\circ g$

 (b) $g\circ f$

(c) $f\circ f$

 (d)  $g\circ g$

State the domain of each composite function.    

$$\begin{gathered} f\left(x\right)=\sqrt{x-2} \\ g\left(x\right)=1-2x \end{gathered}$$

In Problems 29 – 44, for the given functions f and g, find$\left(a\right) f\circ g \left(b\right) g\circ f \left(c\right) f\circ f \left(d\right) g\circ g$  

State the domain of each composite function.

$f\left(x\right)=x^2+1; g\left(x\right)=\sqrt{x-1}$

In Problems 29 – 44, for the given functions f and g, find$\left(a\right) f\circ g \left(b\right) g\circ f \left(c\right) f\circ f \left(d\right) g\circ g$

State the domain of each composite function.

$f\left(x\right)=x^2+4; g\left(x\right)=\sqrt{x-2}$

In Problems 29 – 44, for the given functions f and g, find: (a) $f\circ g$ (b) $g\circ f$ (c) $f\circ f$

 (d) $g\circ g$

State the domain of each composite function.

$f\left(x\right)=\frac{x-5}{x+1},g\left(x\right)=\frac{x+2}{x-3}$

In Problems 29 – 44, for the given functions f and g, find: $\left(a\right) f\circ g \left(b\right) g\circ f \left(c\right) f\circ f \left(d\right) g\circ g$

$f\left(x\right)=\frac{2x-1}{x-2},g\left(x\right)=\frac{x+4}{2x-5}$