Exponential Functions

Exponential Functions

Exponential Functions, Equations and Inequalities

Multiple Select

Which of the following is/are exponential equation/s?

• $32^{\left(x-4\right)}\le16^{\left(x+2\right)}$

• $36^x=6$

• $f\left(x\right)=x^3$

• $\left(\frac{1}{2}\right)^{x+2}=\left(\frac{1}{8}\right)^x$

• $27<3^{x+2}$

Multiple Select

Which of the following is/are exponential inequality/ies?

• $32^{\left(x-4\right)}\le16^{\left(x+2\right)}$

• $36^x=6$

• $f\left(x\right)=x^3$

• $\left(\frac{1}{2}\right)^{x+2}=\left(\frac{1}{8}\right)^x$

• $27<3^{x+2}$

Multiple Select

Which of the following is/are exponential function/s?

• $32^{\left(x-4\right)}\le16^{\left(x+2\right)}$

• $36^x=6$

• $f\left(x\right)=x^3$

•  $f\left(x\right)=\left(\frac{1}{2}\right)^{x+2}$ 

•  $y=2\cdot3^{x+2}$ 

Solving Exponential Equations

• If b > 0 such that b ≠ 1, then b^x = b^y, if and only if x = y, for any real numbers x and y.

Multiple Choice

What should be considered in solving an exponential equation?

• Bases on both sides must be the same.

• Bases on both sides must be simplified.

• Exponents on both sides must be the same.

• Exponents on both sides must be simplified.

Multiple Choice

In solving for the value of the unknown variable in 4 ^𝑥+1 = 16, what is the best thing to do first?

• Divide 16 by 4.

• Multiply 4 by x+1.

• Express 16 as 4².

• Write x+1 as the exponent for both 4 and 16.

Multiple Choice

What is the value of x in 4^𝑥+1 = 16?

• 0

• 1

• 2

• 4

Observe each of the following pairs of solutions. Who got the correct answer? Choices are on the next slide.

Multiple Choice

From the given pairs of solutions, who got the correct answer?

• Becca

• Celia

• Becca and Celia were correct.

• Becca and Celia were incorrect.

Multiple Choice

What is the value of x in 2^5𝑥 = 64?

• $\frac{5}{6}$

• $\frac{6}{5}$

• 6

• 8

Solving Exponential Inequalities

• If b > 1, then b^x > b^y, if and only if x > y.

• If b > 1, then b^x < b^y, if and only if x < y.

• If 0 < b < 1, then b^x > b^y, if and only if x < y.

• If 0 < b < 1, then b^x < b^y, if and only if x > y.

• It is also true for  $\ge and\ \le$  . 

Multiple Choice

Which of the following is equivalent to 10^𝑥−5 > 100^𝑥−10?

• 10(10)^𝑥−5 > 100^𝑥−10

• 10^𝑥−5 > 10^2𝑥−20

• 10^𝑥−5 < 10^2𝑥−20

• 10(10)^𝑥−5 < 100^𝑥−10

Multiple Choice

What is an important step in solving for x in any exponential inequality?

• Consider if 𝑏 > 1 or if 0 < 𝑏 < 1.

• Assume that 𝑏 < 0.

• See to it that the exponents are equal.

• Always divide both sides by the common exponent.

Multiple Choice

Which among the following is a significant observation when solving for x value of 25^𝑥 < 125^𝑥−3?

• The exponents are almost the same.

• The exponents both use x variable.

• The bases are greater than 1.

• The bases are both multiples of 5.

Multiple Choice

What is the interval notation represents the exponential inequality 25^𝑥 < 125^𝑥−3?

• ( 9 , ∞ )

• [ 9 , ∞ )

• ( - ∞ , 9 ]

• ( - ∞ , 9 )

Multiple Choice

Which of the following is the solution of 0.49^𝑥 > 0.7^𝑥+1?

• x < 1

• x > 1

• x > -1

• x < -1

Observe each of the following pairs of solutions. Who got the correct answer? Choices are on the next slide.

Multiple Choice

From the given pairs of solutions, who got the correct answer?

• Dindo

• Hector

• Dindo and Hector were correct.

• Dindo and Hector were incorrect.

Graphing Exponential Functions

Multiple Choice

What is the domain of the given graph?

• $\left(-\infty,\ \infty\right)$

• $\left(0,\infty\right)$

• $\left(-\infty,0\right)$

Multiple Choice

Which of the following statement is always true?

• For any given x-value, the y-value of 𝑦 = 2^𝑥 is negative.

• For any given x-value, the y-value of 𝑦 = 2^𝑥 is all real numbers.

• The domain of exponential function 𝑦 = 2^𝑥 is positive numbers.

• The domain of exponential function 𝑦 = 2^𝑥 is all real numbers.

Multiple Choice

What is the range of the exponential function 𝑓(𝑥) = −3 ∙ 2^𝑥 + 4?

• {𝑦 | 𝑦 ∈ ℝ }

• {𝑦 | 𝑦 ∈ ℝ, 𝑦 > 4}

• . {𝑦 | 𝑦 ∈ ℝ, 𝑦 < 4}

Multiple Choice

Where do the graphs of y = a^x and y = a^-x intersect?

• They intersect at the point (0,0)

• They intersect at the point (1,0)

• They intersect at the point (0,1)

• They intersect at the point (1,1)

Multiple Choice

What will you find if zero is substituted to x-variable of an exponential function?

• asymptote

• y-intercept

• zero

• domain

Multiple Choice

Which determines the equation of the asymptote in 𝑓(𝑥) = 𝑎(𝑏^𝑥−𝑐 ) + 𝑑?

• a

• b

• c

• d

Multiple Choice

Which is/are similar among 𝑓(𝑥) = 2^𝑥 , 𝑔(𝑥) = 4^𝑥 𝑎𝑛𝑑 ℎ(𝑥) = 7^𝑥 ?

• asymptotes

• y-intercepts

• both a and b

• none

Multiple Choice

What is the asymptote of 𝑔(𝑥) = 2^𝑥 + 7?

• 2

• 7

• y = 2

• y = 7

Graphing Exponential Functions

Adding different parameters to a function affects it graph. The graph of the exponential function can be shrunk, stretched, shifted vertically or horizontally, and reflected about the x- and y- axes.

Graphing Exponential Functions

1.Reflection:

a. The graph of y=-b^x is the reflection about the x-axis of the graph of y=b^x .

b. The graph of y=b^-x is the reflection about the y-axis of the graph of y=b^x .

Graphing Exponential Functions

2. Vertical Stretch/Shrink:

Let a > 0 , meaning a is positive.

The y-coordinate of each point in the graph of y=a(b^x ) is equal to the y-coordinate of each point of multiplied by a.

  a. When a > 1, the graph of y=b^x is stretched  vertically.

  b. When 0 < a < 1, the graph of y=b^x is shrunk  vertically.

Note: The y-intercept of y=a(b^x ) is ( 0 , a ) .

Graphing Exponential Functions

3. Vertical Shifts:

Let d be a real number.

The y-coordinate of each point in y=b^x +d is equal to the y-coordinate of each point in y=b^x increased by d .

  a. If d > 0 , meaning d is positive, the graph of y=b^x shifts d units upward.

  b. If d < 0 , meaning d is negative, the graph of y=b^x shifts d units downward.

Note: The y-intercept and the horizontal asymptote of y=b^x will also shift accordingly when d is added.

Graphing Exponential Functions

4. Horizontal Shifts:

Let c be a real number.

The x-coordinate of each point in y=b^x-c is equal to the x-coordinate of each point in y=b^x decreased by c.

  a. If c > 0 , the graph of y=b^x shifts c units to the right.

  b. If c < 0 , the graph of y=b^x shifts c units to the left.

Note: the y-intercept of y=b^x will also shift accordingly when c is added .(c is the value of x )

Multiple Select

Given the function $f\left(x\right)=\left(\frac{1}{3}\right)^{x+2}+1$  , what transformation/s will the graph has?

• reflection

• vertical translation

• horizontal translation

• vertical stretch/shrink

Multiple Select

Given the function $f\left(x\right)=\left(\frac{2}{3}\right)^x-5$  , what transformation/s will the graph has?

• reflection

• vertical translation

• horizontal translation

• vertical stretch/shrink

Multiple Choice

Given the graph of parent function f(x) and the transformed graph g(x), what is the equation of g(x)?

• $g\left(x\right)=3^{x-4}+5$

• $g\left(x\right)=3^{x-4}-5$

• $g\left(x\right)=3^{x+4}-5$

• $g\left(x\right)=3^{x+4}+5$