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Properties of Functions

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

Even and Odd Functions

2 Activities

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

Inverse Functions

3 Activities

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

Exponential Functions

3 Activities


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.
iii. Exactly one of the functions has an inverse.
More Even and Odd Functions Questions:
Combining Families of Functions
Practice: Combining Families of Functions
Is the function f(x)=x2x2−6f(x)=\dfrac{x^2}{x^2-6} even, odd, or neither?
Combining Families of Functions
Practice: Combining Families of Functions
Is the function f(x)=xx2−1f(x)=x\sqrt{x^2-1} even, odd, or neither?
Functions & Graph Properties
Practice: Functions & Graph Properties
True or False:
The function y=−2x3+5y=-2x^3+5 is odd.
Functions & Graph Properties
Practice: Functions & Graph Properties
True or False:
The function y=xx2−1y=\dfrac{x}{x^2-1} is odd.
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.
More Inverse Functions Questions:
Finding an Inverse
Practice Question: Finding an Inverse
Find the inverse of f(x)=x+32−5xf\left(x\right)=\frac{x+3}{2-5x}.
Finding the Inverse of a Function
Find the domain of the function f(x)=x+2x−1f\left(x\right)=\frac{x+2}{x-1}, and a formula for its inverse.
Inverse Function Values
If f(−12)=ef\left(-\frac{1}{2}\right)=e, and ff is a one-to-one function, then we know that
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
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
Inverse and Logarithmic Functions
Find the inverse function of 𝑓(𝑥) = ln (𝑥2 − 5) on its appropriate domain.
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
Inverse Functions
Additional Practice Problems
For each of the following functions, find its inverse and state the domain and range:
a. f(x)=−2(x−1)2+3,   (x≥1)f\left(x\right)=-2\left(x-1\right)^2+3,\ \ \ \left(x\ge1\right)
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
Finding an inverse function
Find the inverse function of 𝑓(𝑥) = 2ln (𝑥 + 𝑒).
Finding an Inverse
Practice Question: Finding an Inverse
Find the inverse of f(x)=x+32−5xf\left(x\right)=\frac{x+3}{2-5x}.
Finding the Inverse of a Function
Find the domain of the function f(x)=x+2x−1f\left(x\right)=\frac{x+2}{x-1}, and a formula for its inverse.
Inverse Function Values
If f(−12)=ef\left(-\frac{1}{2}\right)=e, and ff is a one-to-one function, then we know that
Inverse Functions
Let f(x)=3xx+4\displaystyle f(x)=\frac{3x}{x+4}. Determine f−1(x)f^{-1}(x).
Inverse Functions & Transformations
Practice: Inverse Functions & Transformations
Let y=f(x) y=f(x)~ be graphed below:
Find and graph f−1(x)f^{-1}(x)
Inverse Functions & Transformations
Practice: Inverse Functions & Transformations
Let y=f(x) y=f(x)~ be graphed below:
Find and graph f−1(x)f^{-1}(x)
Inverse Functions
Practice: Inverse Functions
If f−1(x)=−8x3x−5,f^{-1}(x)=\dfrac{-8x}{3x-5}, then determine f(x)f(x).
Inverse Functions
Practice: Inverse Functions
If f−1(x)=x+42x−3−1,f^{-1}(x)=\dfrac{x+4}{2x-3}-1, then determine f(x)f(x).
Inverse Functions
Practice: Inverse Functions
Let f(x)=5−4x+6f(x)=5-\dfrac{4}{x+6}. Find f−1(x)f^{-1}(x).
Inverse Functions
Practice: Inverse Functions
Let f(x)=32x+1f(x)=\dfrac{3}{2x+1}. Find f−1(x)f^{-1}(x).
Inverse Functions
Practice: Inverse Functions
True or False:
The inverse of the function 5x+8y−13=05x+8y-13=0 is f−1(x)=−85x+135f^{-1}(x)=-\dfrac{8}{5}x+\dfrac{13}{5}
Inverse Functions
Practice: Inverse Functions
True or False:
The inverse of the function y=−25(x+3)2+2y=-\frac{2}{5}\left(x+3\right)^2+2 is f−1(x)=3±5−5x2f^{-1}(x)=3\pm\sqrt{5-\dfrac{5x}{2}}
Inverse Functions
Which of the following functions have an inverse?
i. 𝑓(𝑥) = 3𝑥 − 1
ii. 𝑓(𝑥) = −5
Inverse Functions
Find the inverse of the function f(x)=x−13−2xf\left(x\right)=\frac{x-1}{3-2x}.
Practice: Inverse function
Find the inverse of y=x+1+2y=\sqrt{x+1}+2 and its domain.
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
Inverse and Logarithmic Functions
Find the inverse function of 𝑓(𝑥) = ln (𝑥2 − 5) on its appropriate domain.
Finding the Inverse of a Function
Practice: Finding the Inverse of a Function
Find the inverse f−1f^{-1} of the function f(x)=2−e3x+1f\left(x\right)=2-e^{3x+1}, and determine its domain.
Finding an Inverse
Practice Question: Finding an Inverse
Find the inverse of f(x)=x+32−5xf\left(x\right)=\frac{x+3}{2-5x}.
Finding an inverse function
Find the inverse function of 𝑓(𝑥) = 2ln (𝑥 + 𝑒).
Inverse Functions
Additional Practice Problems
For each of the following functions, find its inverse and state the domain and range:
a. f(x)=−2(x−1)2+3,   (x≥1)f\left(x\right)=-2\left(x-1\right)^2+3,\ \ \ \left(x\ge1\right)
Inverse Function Values
If f(−12)=ef\left(-\frac{1}{2}\right)=e, and ff is a one-to-one function, then we know that
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.
Finding the Inverse of a Function
Find the domain of the function f(x)=x+2x−1f\left(x\right)=\frac{x+2}{x-1}, and a formula for its inverse.
More Exponential Functions Questions:
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}
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