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Advanced Geometry : How to graph an exponential function

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Example Question #41 : Coordinate Geometry

Give the -coordinate of the -intercept of the graph of the function

$f(x) = 2 \cdot 10 ^{x} - 8$

Possible Answers:

10,000

$\log 4$

Correct answer:

Explanation:

The -intercept of the graph of f(x) is the point at which it intersects the -axis. Its -coordinate is 0; its -coordinate is f(0) , which can be found by substituting 0 for in the definition:

$f(x) = 2 \cdot 10 ^{x} - 8$

$f(0) = 2 \cdot 10 ^{0} - 8 = 2 \cdot 1 - 8 = 2 - 8 = -6$

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Example Question #41 : Coordinate Geometry

Give the -coordinate of the -intercept of the graph of the function $f(x) = 2e^{x}+ 7$ .

Possible Answers:

The graph of f(x) has no -intercept.

2e+7

$\ln \frac{7}{2}$

Correct answer:

Explanation:

The -intercept of the graph of f(x) is the point at which it intersects the -axis. Its -coordinate is 0; its -coordinate is f(0) , which can be found by substituting 0 for in the definition:

$f(x) = 2e^{x}+ 7$

$f(0) = 2e^{0}+ 7 = 2 \cdot 1 + 7 = 2+7 = 9$ ,

the correct choice.

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Example Question #41 : Coordinate Geometry

Give the -coordinate of the -intercept of the graph of the function $f(x) = 2e^{x}+ 7$ .

Possible Answers:

$\frac{7}{2}$

$\ln \frac{7}{2}$

The graph of f(x) has no -intercept.

Correct answer:

The graph of f(x) has no -intercept.

Explanation:

The -intercept(s) of the graph of f(x) are the point(s) at which it intersects the -axis. The -coordinate of each is 0,; their -coordinate(s) are those value(s) of for which f(x) = 0 , so set up, and solve for , the equation:

f(x)= 0

$2e^{x}+ 7 = 0$

Subtract 7 from both sides:

$2e^{x}+ 7 - 7 = 0 - 7$

$2e^{x}= - 7$

Divide both sides by 2:

$\frac{2e^{x}}{2}=\frac{ - 7}{2}$

$e^{x} =-\frac{ 7}{2}$

The next step would normally be to take the natural logarithm of both sides in order to eliminate the exponent. However, the negative number $-\frac{ 7}{2}$ does not have a natural logarithm. Therefore, this equation has no solution, and the graph of f(x) has no -intercept.

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Example Question #141 : Advanced Geometry

Give the -coordinate of the -intercept of the graph of the function

$f(x) = 2 \cdot 10 ^{x} - 8$

Possible Answers:

10,000

$\log 4$

Correct answer:

$\log 4$

Explanation:

f(x)= 0

$2 \cdot 10 ^{x} - 8 = 0$

Add 8 to both sides:

$2 \cdot 10 ^{x} - 8 + 8 = 0 + 8$

$2 \cdot 10 ^{x} = 8$

Divide both sides by 2:

$\frac{2 \cdot 10 ^{x}}{2} =\frac{ 8}{2}$

$10 ^{x} =4$

Take the common logarithm of both sides to eliminate the base:

$\log 10 ^{x} =\log 4$

$x=\log 4$

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Example Question #51 : Coordinate Geometry

Give the domain of the function $f(x) = 2e^{x-3}- 7$ .

Possible Answers:

$\left \{ x|x> -7\right \}$

$\left \{ x|x \ge -7\right \}$

$\left \{ x|x \ge -3\right \}$

The set of all real numbers

$\left \{ x|x> -3\right \}$

Correct answer:

The set of all real numbers

Explanation:

Let $g(x) = e^{x}$ . This function is defined for any real number , so the domain of g(x) is the set of all real numbers. In terms of g(x) ,

f(x) = 2g(x-3) - 7

Since g(x) is defined for all real , so is g(x-3) ; it follows that f(x) is as well. The correct domain is the set of all real numbers.

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Example Question #51 : Coordinate Geometry

Give the range of the function $f(x) = 2e^{x-3}- 7$ .

Possible Answers:

$[-7, \infty )$

$(-3, \infty )$

$[-3, \infty )$

The set of all real numbers

$(-7, \infty )$

Correct answer:

$(-7, \infty )$

Explanation:

Since a positive number raised to any power is equal to a positive number,

$e^{x-3} > 0$

Applying the properties of inequality, we see that

$2 \cdot e^{x-3} > 2 \cdot 0$

$2 e^{x-3} > 0$

$2 e^{x-3} -7 > 0 - 7$

$2 e^{x-3} -7 > - 7$

f(x) > - 7 ,

and the range of f(x) is the set $(-7, \infty )$ .

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Advanced Geometry : How to graph an exponential function

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #41 : Coordinate Geometry

Example Question #41 : Coordinate Geometry

Example Question #41 : Coordinate Geometry

Example Question #141 : Advanced Geometry

Example Question #51 : Coordinate Geometry

Example Question #51 : Coordinate Geometry

All Advanced Geometry Resources

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