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Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Exponential Functions

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Exponential Functions

Exponential Rules

Multiplying PowersDividing PowersPower of Powersax×ay=ax+yaxay=ax−y(ax)y=axyax×bx=(a×b)xaxbx=(ab)x(ax×by)n=axn×byn(axby)n=axnbynFractional ExponentsNegative ExponentsZero Exponentamn=amna−n=1ana0=1a1n=an\begin{array}{|c|c|c|} \hline \colorFour{\text{Multiplying Powers}}&\colorFour{\text{Dividing Powers}}&\colorFour{\text{Power of Powers}}\\ \hline\\ \displaystyle a^x\times a^y=a^{x+y}&\displaystyle \frac{a^x}{a^y}=a^{x-y}&\displaystyle (a^x)^y=a^{xy}\\\\ \displaystyle a^x\times b^x=(a\times b)^x&\displaystyle \frac{a^x}{b^x}=\left(\frac{a}{b}\right)^x&\displaystyle (a^x\times b^y)^n=a^{xn}\times b^{yn}\\\\ &&\displaystyle \left(\frac{a^x}{b^y}\right)^n=\frac{a^{xn}}{b^{yn}}\\\\\hline \colorFour{\text{Fractional Exponents}}&\colorFour{\text{Negative Exponents}}&\colorFour{\text{Zero Exponent}}\\ \hline\\ \displaystyle a^{\frac{m}{n}}=\sqrt[n]{a^m}&\displaystyle a^{-n}=\frac{1}{a^n}&a^0=1\\\\ \displaystyle a^{\frac{1}{n}}=\sqrt[n]{a}\\\\ \hline \end{array}

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Exponential Functions and Graphs

Exponential Functions have the form: f(x)=ax, for a>1f\left(x\right)=a^x , \text{ for } a>1.
f(x)f(x) has a horizontal asymptote at
y=0
.

The most common exponential function is f(x)=exf\left(x\right)=e^x, where ee is Euler's number.

Euler's Number

The exponential number (Euler's number) is e=2.71828...e=2.71828...

Example: Which is larger: 3−23^{-2}, 2−22^{-2},or e−2e^{-2}?
2−2>e−2>3−22^{-2}>e^{-2}>3^{-2}
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Example: Solving Exponential Equations

Solve for xx in 2x2−x=642^{x^2-x}=64

2x2−x=64\displaystyle 2^{x^2-x}=64
2x2−x=262^{x^2-x}=2^6
x2−x=6x^2-x=6
x2−x−6=0x^2-x-6=0
(x−3)(x+2)=0\left(x-3\right)\left(x+2\right)=0
x=3   or   x=−2x=3\ \ \ \text{or}\ \ \ x=-2
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Example: Exponential Functions

What are the domain and range of the following function: f(x)=ex−5+2f(x) = e^{x-5}+2

exe^x has a domain of all real numbers, so f(x)f(x)has a domain of all real numbers, x∈Rx\in\mathbb{R}

The standard graph of y=exy = e^xhas a range ofy>0y>0, andf(x)f(x) is shifted two units up, so the range is y>2y>2.

domain(f)=I ⁣R and range(f)=(2,∞)\text{domain}(f) = \R \: \text{and} \: \text{range}(f) = (2,\infty)
Extra Practice
Transforming an Exponential
Given f(x)=−2e3x−6+1f\left(x\right)=-2e^{3x-6}+1 , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Which of the following functions, if any, has an inverse?
i. f(x)=3x−1f\left(x\right)=3x-1
ii. f(x)=−5f\left(x\right)=-5
Exponential Functions
Solve for xx in the equation 3x2−15=(x2−5)e4−x3x^2−15=(x^2−5)e^{4-x}
Exponential Functions
If 3x2−15=(x2−5)e4−x3x^2−15=(x^2−5)e^{4-x}, find xx
Exponential Functions
Solve for xx in the equation 3x2−15=(x2−5)e4−x3x^2−15=(x^2−5)e^{4-x}
Properties of Functions
Which of the following statements is/are true about the functions f(x)=0f\left(x\right)=0, g(x)=x3 cos⁡xg\left(x\right)=x^3\ \cos x, and h(x)=esin⁡xh\left(x\right)=e^{\sin x}?
i. Exactly two of the functions are even.
ii. Exactly two of the functions are odd.
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Properties of Exponential Functions
If f(x)=−4⋅e3xπ⋅22xf\left(x\right)=\frac{-4\cdot e^{3x}}{\pi\cdot2^{2x}}, which of the following statements is true?
Exponential Functions: Domain & Range
Find the domain and range of f(x)=2−9x−8f\left(x\right)=\sqrt{2^{-9x}-8}.
Exponential Functions
Which of the following statements is/are true about the exponential function f(x)=C4x+bf\left(x\right)=C4^x+b whose graph passes through the points (1,1)\left(1,1\right)and (12,2)\left(\frac{1}{2},2\right)?
i. f(x)=2(4x)−1f\left(x\right)=2\left(4^x\right)-1
ii. It has a horizontal asymptote at y=−1y=-1
Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to y=exy=e^x:
Shifting the graph 2 units down and 1 unit left
Reflecting along the xx axis
Practice: Exponents Equation
Q:\bf{Q:} Solve for xx in 2x2−x=642^{x^2-x}=64
Exponential Functions
Find the domain of the function f(x)=1(ex−1)(x+1)+12−x2f\left(x\right)=\dfrac{1}{\left(e^x-1\right)\left(x+1\right)}+\dfrac{1}{\sqrt{2-x^2}}
Exponential and Trigonometric Functions
Q:\bf{Q:} Find the domain of . f(x)=sin⁡(3x)e4x−1−tan⁡(2x)f\left(x\right)=\dfrac{\sin\left(3x\right)}{e^{4x-1}}-\tan\left(2x\right)
Exponential Functions: Inequalities with Exponents
Find all xx that satisfy the inequality ex>e−xe^x>e^{−x}
Transforming an Exponential
Given f(x)=−2e3x−6+1f\left(x\right)=-2e^{3x-6}+1 , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Exponential Functions
Solve for xx in the equation 3x2−15=(x2−5)e4−x3x^2−15=(x^2−5)e^{4-x}
Solving Exponential Equations
Solve 5(1128)2x+5=10(32)x+15\Bigg(\dfrac{1}{128}\Bigg)^{2x+5}=10(32)^{x+1}
Simplifying Exponential Expressions
Simplify (72ab(2a+b)32a−3b)\Bigg(\dfrac{72^{ab}(2^{a+b})}{3^{2a-3b}}\Bigg)
Solving Exponential Equations
Solve 4(8)3x+1=32(4)x+24(8)^{3x+1}=32(4)^{x+2}
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x: 4x−6(2x)+8=04^x-6(2^x)+8=0
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x: 32x−3x=03^{2x}-3^x=0
Solving Exponential Equations
Practice: Solving Exponential Equations
What value of xx makes the following statement true? Leave answer in exact form.
(216125)3x=(3625)2x−1\Bigg(\dfrac{216}{125}\Bigg)^{3x}=\Bigg(\dfrac{36}{25}\Bigg)^{2x-1}
Solving Exponential Equations
Practice: Solving Exponential Equations
What value of xx makes the following statement true? Leave answer in exact form.
(34)2x−1=(81256)x+2\Bigg(\dfrac{3}{4}\Bigg)^{2x-1}=\Bigg(\dfrac{81}{256}\Bigg)^{x+2}
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x: 254x+5=125x−325^{4x+5}=125^{x-3}. Leave answer as a fraction.
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x: 42x+1=83x−14^{2x+1}=8^{3x-1}. Leave answer as a fraction.
Exponential, Logarithmic and Trigonometric Functions
Which of the following numbers is the largest?
Practice: Transformation of Exponential Function
Practice: Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to y=exy=e^x:
Shifting the graph 2 units down and 1 unit left
Practice: Transformation of Exponential Function
Practice: Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to y=exy=e^x:
Shifting the graph 2 units down and 1 unit left
Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Exponential Functions: Domain & Range
Find the domain and range of f(x)=2−9x−8f\left(x\right)=\sqrt{2^{-9x}-8}.
Properties of Functions
Which of the following statements is/are true about the functions f(x)=0f\left(x\right)=0, g(x)=x3 cos⁡xg\left(x\right)=x^3\ \cos x, and h(x)=esin⁡xh\left(x\right)=e^{\sin x}?
i. Exactly two of the functions are even.
ii. Exactly two of the functions are odd.
Exponential Functions
Find the domain of the function f(x)=e2x−8f\left(x\right)=e^{\sqrt{2x-8}}
Transformations
Which of the following is the equation of the function produced by vertically shifting the graph of y=exy=e^x up 3 units and then vertically stretching it by a factor of 4?
Q:\bf{Q:} The lifespan of a mammal species is related to its heart rate by the function L(h)=120h−0.01L\left(h\right)=120h^{-0.01} where LL is the lifespan in years and hh is the heart rate in beats per minute. Which of the following observations are true? Select all that apply.
Note: This question is for practice purposes only, it is not based on real scientific data.
Which of this functions represents a quantity that doubles every 33 years?
Practice: Solving Exponential Equations
Find all solutions to the equation 42x⋅8x2+x=144^{2x}\cdot8^{x^2+x}=\frac{1}{4}.
Transforming an Exponential
Given f(x)=−2e3x−6+1f\left(x\right)=-2e^{3x-6}+1 , state all the transformations and draw a rough sketch of the function. State the asymptotes, intercepts, domain and range.
Exponential Functions
Solve for xx in the equation 3x2−15=(x2−5)e4−x3x^2−15=(x^2−5)e^{4-x}
Exponential Functions: Inequalities with Exponents
Find all xx that satisfy the inequality ex>e−xe^x>e^{−x}
Exponential and Trigonometric Functions
Q:\bf{Q:} Find the domain of . f(x)=sin⁡(3x)e4x−1−tan⁡(2x)f\left(x\right)=\dfrac{\sin\left(3x\right)}{e^{4x-1}}-\tan\left(2x\right)
Exponential Functions
Find the domain of the function f(x)=1(ex−1)(x+1)+12−x2f\left(x\right)=\dfrac{1}{\left(e^x-1\right)\left(x+1\right)}+\dfrac{1}{\sqrt{2-x^2}}
Practice: Exponents Equation
Q:\bf{Q:} Solve for xx in 2x2−x=642^{x^2-x}=64
Practice: Exponential function
The exponential function y=bx+k, (b>0)y=b^x+k,\ \left(b>0\right) passes through the point (2,1316)\left(2,\frac{13}{16}\right) and has a horizontal asymptote at y=14y=\frac{1}{4}.
Find the equation of this graph.
Exponential Functions
Solve the equation for xx .
e2x=11ex−30e^{2x}=11e^x-30
Exponential Functions
If h(x)=ex+2x+1h\left(x\right)=e^x+2x+1, find the value of h−1(2)h^{-1}\left(2\right)
Exponential Functions
Find the domain of the function f(x)=18−2−3xf\left(x\right)=\frac{1}{\sqrt{8-2^{-3x}}}
Practice: Log & Exponential Function Properties
Practice: Log & Exponential Function Properties
Which of the following functions is/are always increasing?
Exponential Functions
The exponential function y=bx+k, (b>0)y=b^x+k,\ \left(b>0\right) passes through the point (2,1316)\left(2,\frac{13}{16}\right) and has a horizontal asymptote at y=14y=\frac{1}{4}.
Find the equation of this graph.
Which of the following functions, if any, has an inverse?
i. f(x)=3x−1f\left(x\right)=3x-1
ii. f(x)=−5f\left(x\right)=-5
Exponential Functions
Find the domain of the function f(x)=18−2−3xf\left(x\right)=\frac{1}{\sqrt{8-2^{-3x}}}
Properties of Exponential Functions
If f(x)=−4⋅e3xπ⋅22xf\left(x\right)=\frac{-4\cdot e^{3x}}{\pi\cdot2^{2x}}, which of the following statements is true?
Exponential Functions
Which of the following statements is/are true about the exponential function f(x)=C4x+bf\left(x\right)=C4^x+b whose graph passes through the points (1,1)\left(1,1\right)and (12,2)\left(\frac{1}{2},2\right)?
i. f(x)=2(4x)−1f\left(x\right)=2\left(4^x\right)-1
ii. It has a horizontal asymptote at y=−1y=-1
Transformation of Exponential Function
Find the equation of the graph that results from applying the following transformations in order to y=exy=e^x:
Shifting the graph 2 units down and 1 unit left
Reflecting along the xx axis