General Mathematics - First Semester (1st Quarter)
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General Mathematics - First Semester (1st Quarter)

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Questions and Answers

Which of the following represents the cost of hiring guides as a function of the number of tourists who wish to explore the cave?

f(x) = 4/x

Which of the following represents the cost of renting a videoke machine as a piecewise function of the number of days it is rented?

f(x) = 1000 + 400(x-3)

What is the composition of functions f and g, (f∘g)(x)?

(f∘g)(x) = 1/(x+2) - (x-2)/x

What is the composition of functions g and f, (g∘f)(x)?

<p>(g∘f)(x) = 1/x² + √x</p> Signup and view all the answers

What is the composition of functions f and f, (f∘f)(x)?

<p>(f∘f)(x) = 1/(x²+4x+4)²</p> Signup and view all the answers

Which of the following is a valid piecewise function representing the cost of renting a videoke machine for the given scenario?

<p>$f(x) = 1000 + 400(x-3)$</p> Signup and view all the answers

Which of the following is the correct composition of functions f and g, $(f \circ g)(x)$?

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

Which of the following is the correct composition of functions f and f, $(f \circ f)(x)$?

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

Which of the following is a valid piecewise function representing the cost of renting a videoke machine for the given scenario?

<p>$C(x) = 1000$ for $x \leq 3$, $C(x) = 1000 + 400(x-3)$ for $x &gt; 3$</p> Signup and view all the answers

What is the composition of functions g and f, $(g \circ f)(x)$?

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

Which of the following represents the cost of hiring guides as a function of the number of tourists who wish to explore the cave?

<p>$C(x) = 4x$</p> Signup and view all the answers

Which of the following is a characteristic of a rational function?

<p>The function is in the form of f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions and q(x) ≠ 0.</p> Signup and view all the answers

Which of the following is the correct form of a rational function?

<p>f(x) = 2x + 4 / x^2 - 1</p> Signup and view all the answers

What is the domain of a rational function?

<p>The set of real numbers except those values of x that make the denominator zero.</p> Signup and view all the answers

What is an asymptote in a rational function?

<p>A line that approaches the graph but never meets or intersects.</p> Signup and view all the answers

What are intercepts in a rational function?

<p>Points where the graph intersects both the x-axis and the y-axis.</p> Signup and view all the answers

Which of the following is the correct form of a rational function?

<p>$f(x) = \frac{p(x)}{q(x)}$</p> Signup and view all the answers

What is an asymptote in a rational function?

<p>A line that approaches the graph but never meets or intersects.</p> Signup and view all the answers

What is the domain of a rational function?

<p>The set of real numbers except those values of x that will make the denominator zero.</p> Signup and view all the answers

Which of the following is the correct form of a rational function?

<p>$f(x) = \frac{x-4}{x-2}$</p> Signup and view all the answers

What is an asymptote in a rational function?

<p>A line that approaches the graph but never meets or intersects</p> Signup and view all the answers

What is the domain of a rational function?

<p>The set of real numbers except those values of x that will make the denominator zero</p> Signup and view all the answers

Which of the following is the correct equation to find the inverse of the function $f(x) = 1 + x - 2$?

<p>$x - 1 = y - 2$</p> Signup and view all the answers

Which of the following is the correct equation for the inverse of the function $f(x) = \frac{x-2}{x+2}$?

<p>$x = \frac{y-2}{y+2}$</p> Signup and view all the answers

What is the inverse of the function $f(x) = x^2 + x$?

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

What is the inverse of the function $f(x) = 1 - 4 - 3x$?

<p>$f^{-1}(x) = 1 - 4 - 3x$</p> Signup and view all the answers

Which of the following statements about inverse functions is true?

<p>Inverse functions have the same domain and range.</p> Signup and view all the answers

Which of the following is the correct equation to find the inverse of the function $f(x) = rac{x-2}{x+2}$?

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

Which of the following is the correct equation for the inverse of the function $f(x) = 1 + x - 2$?

<p>$f^{-1}(x) = x^2 - 2x + 3$</p> Signup and view all the answers

Which of the following is the correct equation for the inverse of the function $f(x) = 1 - 4 - 3x$?

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

Which of the following is the correct equation for the inverse of the function $f(x) = \frac{x-2},{x+2}$?

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

What is the domain of a rational function?

<p>All real numbers except 0</p> Signup and view all the answers

Which of the following is the correct composition of functions $f$ and $g$, $(f \circ g)(x)$?

<p>$f(g(x)) = 1 - 4 - 3x$</p> Signup and view all the answers

Which of the following is the correct exponential model for the situation?

<p>$A(t) = 100e^{- rac{t}{250}}$</p> Signup and view all the answers

How much of the substance will remain after 500 years?

<p>25g</p> Signup and view all the answers

Which of the following is the correct exponential model for the situation?

<p>$P(t) = 1000(3)^{ rac{t}{80}}$</p> Signup and view all the answers

What is the size of the population after 100 years?

<p>27,000</p> Signup and view all the answers

Which of the following is the correct exponential model for the situation?

<p>$P(t) = 7000(1.12)^{t-2000}$</p> Signup and view all the answers

Which of the following is the correct exponential model for the situation when a substance with a half-life of 250 years initially has 100g?

<p>$A(t) = 100 \cdot 2^{-\frac{t},{250}}$</p> Signup and view all the answers

How much of the substance will remain after 500 years if it initially had 100g and has a half-life of 250 years?

<p>50g</p> Signup and view all the answers

Which of the following is the correct exponential model for the situation when a population starts with 1,000 individuals and triples every 80 years?

<p>$P(t) = 1000 \cdot 3^{\frac{t},{80}}$</p> Signup and view all the answers

Which of the following is the correct exponential model for the situation when a substance with a half-life of 250 years initially has 100g?

<p>$A(t) = 100 \cdot \left(\frac{1},{2}\right)^{\frac{t},{250}}$</p> Signup and view all the answers

How much of the substance will remain after 500 years if it initially had 100g and has a half-life of 250 years?

<p>50g</p> Signup and view all the answers

Which of the following is the correct exponential model for the situation when a population starts with 1,000 individuals and triples every 80 years?

<p>$P(t) = 1000 \cdot 3^{\frac{t},{80}}$</p> Signup and view all the answers

Which of the following is the correct equation for the inverse of the function $f(x) = rac{x-2}{x+2}$?

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

Which of the following is the correct exponential model for the situation?

<p>$y = ab^x$</p> Signup and view all the answers

What is an asymptote in a rational function?

<p>A line that the graph of the function approaches but never touches.</p> Signup and view all the answers

What is the domain of a rational function?

<p>The set of all real numbers except for the values that make the denominator zero.</p> Signup and view all the answers

How much of the substance will remain after 500 years if it initially had 100g and has a half-life of 250 years?

<p>50g</p> Signup and view all the answers

Which of the following is the correct logarithmic form for the exponential equation $5^2 = 25$?

<p>$\log_5 25 = 2$</p> Signup and view all the answers

Which of the following is the correct logarithmic form for the exponential equation $2^3 = 8$?

<p>$\log_2 8 = 3$</p> Signup and view all the answers

Which of the following is the correct logarithmic form for the exponential equation $10^2 = 100$?

<p>$\log_{10} 100 = 2$</p> Signup and view all the answers

Which of the following is the correct logarithmic form for the exponential equation $10^2 = 100$?

<p>$log_{10} 100 = 2$</p> Signup and view all the answers

What is the domain of a logarithmic function?

<p>All positive real numbers</p> Signup and view all the answers

Which of the following is the correct equation for the inverse of the function $f(x) = \frac{1},{x}$?

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

Which equation represents the solution to the logarithmic equation log2(x) = log2(14)?

<p>x = 14</p> Signup and view all the answers

What is the solution to the logarithmic equation logb(x - 30) = logb(x)?

<p>x = 30</p> Signup and view all the answers

What is the solution to the logarithmic equation 2logb(4) + logb(x - 1) = 3?

<p>x = 4</p> Signup and view all the answers

Which inequality represents the solution to the logarithmic inequality log3(2x - 1) > log3(x + 2)?

<p>x &gt; 3</p> Signup and view all the answers

Which equation represents the solution to the logarithmic equation log2(8) = x?

<p>x = 2</p> Signup and view all the answers

Which of the following is the solution to the equation logb x^2 = logb(4x - 4)?

<p>x = 4</p> Signup and view all the answers

Which of the following is the solution to the equation log2(2x - 1) = 2?

<p>x = 3</p> Signup and view all the answers

Which of the following is the solution to the inequality log3(2x - 1) > log3(x + 2)?

<p>x &gt; 3</p> Signup and view all the answers

Which of the following is the solution to the equation $log_2(x) = log_2(14)$?

<p>x = 14</p> Signup and view all the answers

What is the solution to the logarithmic equation $log_b(x - 30) = log_b(x)$?

<p>x = 30</p> Signup and view all the answers

Which of the following is the correct solution to the equation $2log_b 4 + log_b(x - 1) = 3$?

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

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