## Engage NY Eureka Math Algebra 2 Module 3 Lesson 18 Answer Key

### Eureka Math Algebra 2 Module 3 Lesson 18 Opening Exercise Answer Key

Opening Exercise:

Complete the following table of values of the function f(x) = 2^{x}. We want to sketch the graph of y f(x) and then reflect that graph across the diagonal line with equation y = x.

Answer:

On the set of axes below, plot the points from the table and sketch the graph of y = 2^{x}. Next, sketch the diagonal line with equation y = x, and then reflect the graph of y = 2^{x} across the line.

Answer:

### Eureka Math Algebra 2 Module 3 Lesson 18 Exercise Answer Key

Exercises:

Exercise 1.

Complete the following table of values of the function g(x) = log_{2}(x). We want to sketch the graph of y = g(x) and then reflect that graph across the diagonal line with equation y = x.

Answer:

On the set of axes below, plot the points from the table and sketch the graph of y = log_{2}(x). Next, sketch the diagonal line with equation y = x, and then reflect the graph of y = log_{2}(x) across the line.

Answer:

Exercise 2.

Working independently, predict the relation between the graphs of the functions f(x) = 3^{x} and g(x) = log_{3}(x). Test your predictions by sketching the graphs of these two functions. Write your prediction in your notebook, provide justification for your prediction, and compare your prediction with that of your neighbor.

Answer:

The graphs will be reflections of each other about the diagonal.

Exercise 3.

Now letâ€™s compare the graphs of the functions f_{2}(x) = 2 and f_{3}(x) = 3. Sketch the graphs of the two exponential functions on the same set of axes; then, answer the questions below.

Answer:

a. Where do the two graphs intersect?

Answer:

The two graphs intersect at the point (0, 1).

b. For which values of x is 2^{x} < 3^{x}?

Answer:

If x > 0, then 2^{x} < 3^{x}

c. For which values of x is 2^{x} > 3^{x}?

Aans:

If x < 0, then 2^{x} > 3^{x}

d. What happens to the values of the functions 12 and 13 as x â†’ âˆž?

Answer:

As x â†’ âˆž, both f_{2}(x) â†’ âˆžâ€š and f_{3}(x) â†’ âˆž.

e. What happens to the values of the functions 12 and 13 as x â†’ -âˆž?

Answer:

As x â†’ -âˆž,both f_{2}(x) â†’ 0 and f_{3}(x) â†’ 0.

f. Does either graph ever intersect the x-axis? Explain how you know.

Answer:

No. For every value of x, we know 2^{x} â‰ 0 and 3^{x} â‰ 0.

Exercise 4.

Add sketches of the two logarithmic functions g_{2}(x) = log_{2}(x) and g_{3}(x) = log_{3}(x) to the axes with the graphs of the exponential functions from Exercise 3; then, answer the questions below.

a. Where do the two logarithmic graphs intersect?

Answer:

The two graphs intersect at the point (1, 0).

b. For which values of x is log_{2}(x) < log_{3}(x)?

Answer:

If x < 1, then log_{2}(x) < log_{3}(x).

c. For which values of x is log_{2}(x) > log_{3}(x)?

Answer:

If x > 1, then log_{2}(x) > log_{3}(x).

d. What happens to the values of the functions g_{2} and g_{3} as x â†’ âˆž?

Answer:

As x â†’ âˆž, both g_{2}(x) â†’ âˆž and g_{3}(x) â†’ âˆž.

e. What happens to the values of the functions g_{2} and g_{3} as x â†’ 0?

Answer:

As x â†’ 0, both g_{2}(x) â†’ -âˆž and g_{3}(x) â†’ -âˆž.

f. Does either graph ever intersect the y-axis? Explain how you know.

Answer:

No. Logarithms are only defined for positive values of x.

g. Describe the similarities and differences in the behavior of f_{2}(x) and g_{2}(x) as x â†’ âˆž.

Answer:

As x â†’ âˆž, both f_{2}(x) â†’ âˆž and g_{2}(x) â†’ âˆž; however, the exponential function gets very large very quickly, and the logarithmic function gets large rather slowly.

### Eureka Math Algebra 2 Module 3 Lesson 18 Problem Set Answer Key

Question 1.

Sketch the graphs of the functions f(x) = 5^{x} and g(x) = log_{5}(x).

Answer:

Question 2.

Sketch the graphs of the functions f(x) = (\(\frac{1}{2}\))^{x} and g(x) = log_{\(\frac{1}{2}\)}(x).

Answer:

Question 3.

Sketch the graphs of the functions f_{1}(x) = (\(\frac{1}{2}\))^{x} and f_{2}(x) = (\(\frac{3}{4}\))^{x} on the same sheet of graph paper, and answer the following questions.

a. Where do the two exponential graphs intersect?

Answer:

The graphs intersect at the point (0, 1).

b. For which values of x is (\(\frac{1}{2}\))^{x} < (\(\frac{3}{4}\))^{x}?

Answer:

If x > 0, then (\(\frac{1}{2}\))^{x} < (\(\frac{3}{4}\))^{x}.

c. For which values of x is (\(\frac{1}{2}\))^{x} > (\(\frac{3}{4}\))^{x}??

Answer:

If x < 0, then (\(\frac{1}{2}\))^{x} < (\(\frac{3}{4}\))^{x}.

d. What happens to the values of the functions f_{1} and f_{2} as x â†’ âˆž?

Answer:

As x â†’ âˆž,both f_{1}(x) â†’ 0 and f_{2}(x) â†’ 0.

e. What are the domains of the two functions f_{1} and f_{2}?

Answer:

Both functions have domain (-âˆž, âˆž).

Question 4.

Use the information from Problem 3 together with the relationship between graphs of exponential and logarithmic functions to sketch the graphs of the functions g_{1}(x) = log_{\(\frac{1}{2}\)}(x) and g_{2}(x) = log_{>\(\frac{3}{4}\)</}(x) on the same sheet of graph paper. Then, answer the following questions.

a. Where do the two logarithmic graphs intersect?

Answer:

The graphs intersect at the point (1, 0).

b. For which values of x is log_{\(\frac{1}{2}\)}(x) < log_{\(\frac{3}{4}\)}(x)?

Answer:

When x < 1, we have log_{\(\frac{1}{2}\)}(x) < log_{\(\frac{3}{4}\)}(x).

c. For which values of x is log_{\(\frac{1}{2}\)}(x) > log_{\(\frac{3}{4}\)}(x)?

Answer:

When x > 1, we have log_{\(\frac{1}{2}\)}(x) > log_{\(\frac{3}{4}\)}(x).

d. What happens to the values of the functions g_{1} and g_{2} as x â†’ âˆž?

Answer:

As x â†’ âˆž, both g_{1}(x) â†’ -âˆž and g_{2}(x) â†’ -âˆž.

e. What are the domains of the two functions g_{1} and g_{2}?

Answer:

Both functions have domain (0, âˆž).

Question 5.

For each function f, find a formula for the function h in terms of x.

a. If f(x) = x^{3}, find h(x) = 128f(\(\frac{1}{4}\)x) + f(2x).

Answer:

h(x) = 10x^{3}

b. If (x) = x^{2} + 1, find h(x) = f(x + 2) – f(2).

Answer:

h(x) = x^{2} + 4x

c. If f(x) = x^{3} + 2x^{2} + 5x + 1, find h(x) = \(\frac{f(x)+f(-x)}{2}\)

Answer:

h(x) = 2x^{2} + 1

d. If (x) = x^{3} + 2x^{2} + 5x + 1, find h(x) = \(\frac{f(x)-f(-x)}{2}\)

Answer:

h(x) = x^{3} + 5x

Question 6.

In Problem 5, parts (c) and (d), list at least two aspects about the formulas you found as they relate to the function f(x) = x^{3} + 2x^{2} + 5x + 1.

Answer:

The formula for 1(c) is all of the even power terms off. The formula for 1(d) is all of the odd power terms off. The sum of the two functions gives f back again; that is, \(\frac{f(x)+f(-x)}{2}\) + \(\frac{f(x)-f(-x)}{2}\) = f

Question 7.

For each of the functions f and g below, write an expression for (i) f(g(x)), (ii) g(f(x)) and (iii) f(f(x)) in terms of x.

a. f(x) = x^{\(\frac{2}{3}\)}, g(x) = x^{12}

Answer:

i. f(g(x)) = x^{8}

ii. g(f(x)) = x^{8}

iii. f(f(x)) = x^{\(\frac{4}{9}\)}

b. f(x) = \(\frac{\boldsymbol{b}}{x-a}\), g(x) = \(\frac{\boldsymbol{b}}{\boldsymbol{x}}\) + a for two numbers a and b, when x is not 0 or a.

Answer:

i. f(g(x)) = x

ii. g(f(x)) = x

iii. f(f(x)) = \(\frac{b}{\frac{b}{x-a}-a}\), which is equivalent to f(f(x)) = \(\frac{b(x-a)}{b+a^{2}-a x}\)

c. f(x) = \(\frac{x+1}{x-1}\) g(x) = \(\frac{x+1}{x-1}\) when x is not 1 or -1

Answer:

i. f(g(x)) = x

ii. g(f(x)) = x

iii. f(f(x)) = x

d. f(x) = 2^{x}, 9(x) = log_{2}(x)

Answer:

i. f(g(x)) = x

ii. g(f(x)) = x

iii. f(f(x)) = x

e. f(x) = ln(x), g(x) = e^{x}

Answer:

i. f(g(x)) = x

ii. g(f(x)) = x

iii. f(f(x)) = ln(ln(x))

f. f(x) = 2 . 100^{x}, g(x) = \(\frac{1}{2}\)log(\(\frac{1}{2}\)x)

Answer:

i. f(g(x)) = x

ii. g(f(x)) = x

iii. f(f(x)) = 2 . 10000^{100x}

### Eureka Math Algebra 2 Module 3 Lesson 18 Exit Ticket Answer Key

Question 1.

The graph of a logarithmic function g(x) = log_{b}(x) is shown below.

a. Explain how to find points on the graph of the function f(x) = b^{x}

Answer:

A point (x, y) is on the graph of f if the corresponding point (y, x) is on the graph of g.

b. Sketch the graph of the function f(x) = b^{x} on the same axes.

Answer: