Calculus I - Functions (Practice Problems) (2024)

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Section 1.1 : Functions

For problems 1 – 4 the given functions perform the indicated function evaluations.

1. \(f\left( x \right) = 3 - 5x - 2{x^2} \) Solution
   1. \(f\left( 4 \right) \)
   2. \(f\left( 0 \right)\)
   3. \(f\left( { - 3} \right) \)
   1. \(f\left( {6 - t} \right) \)
   2. \(f\left( {7 - 4x} \right)\)
   3. \(f\left( {x + h} \right) \)
2. \(\displaystyle g\left( t \right) = \frac{t}{{2t + 6}} \) Solution
   1. \(g\left( 0 \right) \)
   2. \(g\left( { - 3} \right)\)
   3. \(g\left( {10} \right) \)
   1. \(g\left( {{x^2}} \right) \)
   2. \(g\left( {t + h} \right)\)
   3. \(g\left( {{t^2} - 3t + 1} \right) \)
3. \(h\left( z \right) = \sqrt {1 - {z^2}} \) Solution
   1. \(h\left( 0 \right) \)
   2. \(h\left( { - \frac{1}{2}} \right)\)
   3. \(h\left( {\frac{1}{2}} \right) \)
   1. \(h\left( {9z} \right) \)
   2. \(h\left( {{z^2} - 2z} \right) \)
   3. \(h\left( {z + k} \right) \)
4. \(\displaystyle R\left( x \right) = \sqrt {3 + x} - \frac{4}{{x + 1}} \) Solution
   1. \(R\left( 0 \right) \)
   2. \(R\left( 6 \right)\)
   3. \(R\left( { - 9} \right) \)
   1. \(R\left( {x + 1} \right)\)
   2. \(R\left( {{x^4} - 3} \right)\)
   3. \(R\left( {\frac{1}{x} - 1} \right) \)

The difference quotient of a function \(f\left( x \right) \) is defined to be,

\[\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\]

For problems 5 – 9 compute the difference quotient of the given function.

1. \(f\left( x \right) = 4x - 9 \) Solution
2. \(g\left( x \right) = 6 - {x^2} \) Solution
3. \(f\left( t \right) = 2{t^2} - 3t + 9 \) Solution
4. \(\displaystyle y\left( z \right) = \frac{1}{{z + 2}} \) Solution
5. \(\displaystyle A\left( t \right) = \frac{{2t}}{{3 - t}} \) Solution

For problems 10 – 17 determine all the roots of the given function.

1. \(f\left( x \right) = {x^5} - 4{x^4} - 32{x^3} \) Solution
2. \(R\left( y \right) = 12{y^2} + 11y - 5 \) Solution
3. \(h\left( t \right) = 18 - 3t - 2{t^2} \) Solution
4. \(g\left( x \right) = {x^3} + 7{x^2} - x \) Solution
5. \(W\left( x \right) = {x^4} + 6{x^2} - 27 \) Solution
6. \(f\left( t \right) = {t^{\frac{5}{3}}} - 7{t^{\frac{4}{3}}} - 8t \) Solution
7. \(\displaystyle h\left( z \right) = \frac{z}{{z - 5}} - \frac{4}{{z - 8}} \) Solution
8. \(\displaystyle g\left( w \right) = \frac{{2w}}{{w + 1}} + \frac{{w - 4}}{{2w - 3}} \) Solution

For problems 18 – 22 find the domain and range of the given function.

1. \(Y\left( t \right) = 3{t^2} - 2t + 1 \) Solution
2. \(g\left( z \right) = - {z^2} - 4z + 7 \) Solution
3. \(f\left( z \right) = 2 + \sqrt {{z^2} + 1} \) Solution
4. \(h\left( y \right) = - 3\sqrt {14 + 3y} \) Solution
5. \(M\left( x \right) = 5 - \left| {x + 8} \right| \) Solution

For problems 23 – 32 find the domain of the given function.

1. \(\displaystyle f\left( w \right) = \frac{{{w^3} - 3w + 1}}{{12w - 7}} \) Solution
2. \(\displaystyle R\left( z \right) = \frac{5}{{{z^3} + 10{z^2} + 9z}} \) Solution
3. \(\displaystyle g\left( t \right) = \frac{{6t - {t^3}}}{{7 - t - 4{t^2}}} \) Solution
4. \(g\left( x \right) = \sqrt {25 - {x^2}} \) Solution
5. \(h\left( x \right) = \sqrt {{x^4} - {x^3} - 20{x^2}} \) Solution
6. \(\displaystyle P\left( t \right) = \frac{{5t + 1}}{{\sqrt {{t^3} - {t^2} - 8t} }} \) Solution
7. \(f\left( z \right) = \sqrt {z - 1} + \sqrt {z + 6} \) Solution
8. \(\displaystyle h\left( y \right) = \sqrt {2y + 9} - \frac{1}{{\sqrt {2 - y} }} \) Solution
9. \(\displaystyle A\left( x \right) = \frac{4}{{x - 9}} - \sqrt {{x^2} - 36} \) Solution
10. \(Q\left( y \right) = \sqrt {{y^2} + 1} - \sqrt[3]{{1 - y}} \) Solution

For problems 33 – 36 compute \(\left( {f \circ g} \right)\left( x \right) \) and \(\left( {g \circ f} \right)\left( x \right) \) for each of the given pair of functions.

1. \(f\left( x \right) = 4x - 1 \), \(g\left( x \right) = \sqrt {6 + 7x} \) Solution
2. \(f\left( x \right) = 5x + 2 \), \(g\left( x \right) = {x^2} - 14x \) Solution
3. \(f\left( x \right) = {x^2} - 2x + 1 \), \(g\left( x \right) = 8 - 3{x^2} \) Solution
4. \(f\left( x \right) = {x^2} + 3 \), \(g\left( x \right) = \sqrt {5 + {x^2}} \) Solution