Inverse Functions and Exponents Quiz
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Questions and Answers

Which of the following functions does NOT exhibit exponential decay?

  • $i(x) = 2^x$ (correct)
  • $f(x) = 3x$ (correct)
  • $h(x) = 2^{-x}$
  • $g(x) = \frac{1}{4x}$

    • How can the equation $y = \log_3 x$ be expressed in exponential form?

  • $y^3 = x$
  • $3^y = x$ (correct)
  • $y = 3^x$
  • $x^3 = y$

    • What is the value of $\log_5 625$?

  • 4 (correct)
  • 3
  • 6
  • 5

    • If $\log_2 64 = x$, what is the value of x?

    <p>6</p> Signup and view all the answers

    Which of the following logarithmic equations is correctly transformed into exponential form?

    <p>$\log_{10} 0.01 = -2$ becomes $10^{-2} = 0.01$</p> Signup and view all the answers

    What is the result of applying the division rule to the expression $a^{10} / a^{6}$?

    <p>$a^{4}$</p> Signup and view all the answers

    When applying the power of a power rule to $a^{7^{2}}$, what is the correct simplification?

    <p>$a^{14}$</p> Signup and view all the answers

    What is the outcome of the expression $a^{0}$ regardless of the value of $a$?

    <p>$1$</p> Signup and view all the answers

    How would you express the negative exponent $a^{-7}$ in a different form?

    <p>$1/a^{7}$</p> Signup and view all the answers

    Which rule applies to the expression $(ab)^{4}$ to distribute the exponents?

    <p>Power of a product rule</p> Signup and view all the answers

    What is the simplified form of $a^{3}/b^{3}$ using the power of a fraction rule?

    <p>$a^{3}/b^{3}$</p> Signup and view all the answers

    For the fractional exponent $a^{3/5}$, how would it be expressed in radical form?

    <p>$ oot{5}{a^{3}}$</p> Signup and view all the answers

    In the exponential function $f(x) = b^{x}$, which condition must the base $b$ satisfy?

    <p>$b &gt; 0$ and $b eq 1$</p> Signup and view all the answers

    What is the inverse of the function $f(x) = 2x - 5$?

    <p>$f^{-1}(x) = \frac{x + 5}{2}$</p> Signup and view all the answers

    Which of the following statements correctly describes an inverse function?

    <p>It satisfies $f(f^{-1}(x)) = x$ for all $x$.</p> Signup and view all the answers

    What is the inverse of the function $f(x) = 3x + 6$?

    <p>$f^{-1}(x) = \frac{x - 6}{3}$</p> Signup and view all the answers

    If $f(x) = 2x - 4$, which of the following pairs is not on the graph of $f$?

    <p>(1, -2)</p> Signup and view all the answers

    Using the multiplication rule of exponents, what is the result of $m^7 m^3$?

    <p>$m^{10}$</p> Signup and view all the answers

    What is the correct approach to finding the inverse of a function?

    <p>By interchanging $x$ and $y$ and solving for $y$.</p> Signup and view all the answers

    How do you determine if two functions are inverses of each other?

    <p>If $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$</p> Signup and view all the answers

    Which equation represents the function for $f(x) = 3x + 6$ on the graph?

    <p>$y = 3x + 6$</p> Signup and view all the answers

    What is an exponential equation?

    <p>An equation where the base is a constant and the exponent is a variable.</p> Signup and view all the answers

    What form does an exponential growth function take?

    <p>f(x) = ka^x where a &gt; 1 and k &gt; 0</p> Signup and view all the answers

    Which of the following describes exponential decay?

    <p>The half-life of a radioactive substance.</p> Signup and view all the answers

    In the context of exponential functions, what does the constant k represent?

    <p>The initial amount at time zero.</p> Signup and view all the answers

    Which of the following scenarios demonstrates exponential growth?

    <p>Interest accumulating on a savings account.</p> Signup and view all the answers

    What does the power rule for logarithms state regarding expressions of the form $\log_b(M^N)$?

    <p>$N \log_b(M)$</p> Signup and view all the answers

    How does the division rule for logarithms simplify the expression $\log_b(X/Y)$?

    <p>$\log_b(X) - \log_b(Y)$</p> Signup and view all the answers

    Which of the following correctly expands the expression $\log_5(a^2 b^3 c^4)$?

    <p>$2 \log_5(a) + 3 \log_5(b) + 4 \log_5(c)$</p> Signup and view all the answers

    Using the multiplication rule, how can $\log_4(XY)$ be expressed?

    <p>$\log_4(X) + \log_4(Y)$</p> Signup and view all the answers

    What expression results from expanding $\log_3(5^2)$ using the power rule?

    <p>$2 \log_3(5)$</p> Signup and view all the answers

    What is the result when applying the division rule to the expression $\log_5(15) - \log_5(5)$?

    <p>$\log_5(3)$</p> Signup and view all the answers

    When simplifying $3 \log_4(w) + 2 \log_4(x) - 3 \log_4(y) + 4 \log_4(z)$ into a single logarithm, what is the final expression?

    <p>$\log_4(w^3 x^2 / y^3 z^4)$</p> Signup and view all the answers

    What does the expansion of $\log_4(xy^2)$ result in?

    <p>$\log_4(x) + 2 \log_4(y)$</p> Signup and view all the answers

    If $\log_3(A) + \log_3(B)$ simplifies to which expression using the multiplication rule?

    <p>$\log_3(AB)$</p> Signup and view all the answers

    Study Notes

    Inverse Functions

    • An inverse function is a function that reverses the effect of another function.
    • If f is a function, its inverse is denoted by f-1.
    • For a function and its inverse, f(f-1(x)) = x and f-1(f(x)) = x.
    • To find the inverse of a function, switch the x and y variables and solve for y.

    Laws of Exponents

    • Multiplication Rule: ax * ay = ax + y
    • Division Rule: ax / ay = ax - y
    • Power of a Power Rule: (ax)y = axy
    • Power of a Product Rule: (ab)x = ax * bx
    • Power of a Fraction Rule: (a/b)x = ax / bx
    • Zero Exponent: a0 = 1
    • Negative Exponent: a-x = 1 / ax
    • Fractional Exponent: ax/y = √yax

    Exponential Functions

    • An exponential function is a function of the form f(x) = bx where b > 0 and b ≠ 1.
    • To evaluate an exponential function, substitute the value of the variable into the function.

    Exponential Equations

    • Exponential equations are equations where the exponent is a variable.
    • To solve an exponential equation, express both sides of the equation as powers with the same base, then solve the resulting equation.

    Application of Exponential Functions

    • Exponential Growth: Describes a process where a quantity increases at an ever-increasing rate over time.

    • Exponential Decay: Describes a process where a quantity decreases at a consistent percentage rate over time.

    • Examples of Exponential Growth:

      • A colony of bacteria in a human body
      • The population of the Philippines
      • Compound Interest

    • Examples of Exponential Decay:

      • The concentration of medicine in a patient’s body over time
      • The value of gadgets over time
      • The remains of a radioactive element

    Logarithmic Functions

    • A logarithmic function is a function of the form y = logb x where b > 0, b ≠ 1 and x > 0.
    • y = logb x is equivalent to x = by.

    Laws of Logarithms

    • Multiplication Rule: logb (MN) = logb M + logb N
    • Division Rule: logb (M/N) = logb M - logb N
    • Power Rule: logb (MN) = N logb M

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    Description

    Test your understanding of inverse functions and the laws of exponents. This quiz covers key concepts such as how to find inverse functions and the various rules governing exponents. Perfect for students looking to solidify their knowledge in algebra.

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