Math 135 - section 01 - Precalculus Diagnostic Quiz Solutions August 30, 2013

1. Let f(x) = x^{2} − x and g(x) = x + 1. Which function is f(g(x))?

A. x^{2} − x B. x^{2} + 3x C. x^{2} + x D. x^{2} + x + 2 Solution: f(g(x)) = (x + 1)^{2} − (x + 1) = x^{2} + 2x + 1 − x − 1 = x^{2} + x. This is C.

2. Find common factors and cancel to simplify: \frac{12x}{3x − 6} · \frac{x^{2} − 4}{2x + 4}

A. \frac{x^{2} + 12x − 4}{5x − 2} B. 2x C. −\frac{4}{3}(x − 1) D. \frac{12x^{3} − 48x}{6x^{2} − 12}

Solution: Since x^{2} − 4 = (x − 2)(x + 2), 2x + 4 = 2(x + 2), and 3x − 6 = 3(x − 2), after cancelling all common factors, what is left is 2x. The correct answer is B.

3. Which is equal to (u^{−5}v^{2})^{3} (\frac{v^{2}}{u} )^{−1}?

A. u^{−16}v^{8} B. u^{−14}v^{4} C. u^{−9}v^{3} D. (uv)^{−7}

1. Solution: (u^{−5}v^{2})^{3} = u^{−15}v^{6} and (\frac{v^{2}}{u} )^{−1} = uv^{−2}. So adding exponents, we see this is B: u^{−14}v^{4}.

4. If f(x) = 5x^{2} − 11, what is f(a + 1) − f(a)?

Solution: f(a + 1) − f(a) = 5(a + 1)^{2} − 11 − (5a^{2} − 11) = 10a + 5

5. Find all values of x satisfying 2(x − 2) > 5.

Solution: This is the same as 2x > 5 + 4 = 9, so all x > 9/2 are solutions.

6. Solve for x: 2x^{2} − x − 6 = 0 (find all solutions).

Solution: The quadratic equation factors as (2x + 3)(x − 2) = 0, so the solutions are x = −\frac{3}{2} and x = 2.