Mastery Check Solutions

Show What You Know

Determine $f (\frac{1}{2}) if f (- 4) = 40 and f (x) = a x^{2} - 5 x - 12 .$

$\begin{array}{rcl} f (- 4) & = & a {(- 4)}^{2} - 5 (- 4) - 12 \\ 40 & = & 16 a + 20 - 12 \\ 40 & = & 16 a + 8 \\ 32 & = & 16 a \\ a & = & 2 \\ f (x) = 2 x^{2} - 5 x - 12 \end{array}$

$f (\frac{1}{2}) = 2 {(\frac{1}{2})}^{2} - 5 (\frac{1}{2}) - 12 f (\frac{1}{2}) = \frac{1}{2} - \frac{5}{2} - 12 f (\frac{1}{2}) = - 14$

Note

Remember to find $f (\frac{1}{2}) .$ Determining the value of a is the first step, but is not the answer to the problem.

Fill in f(x) based on your work in part A. Then use the functions f(x), g(x), and h(x) to complete parts C, D, and E.

$f (x) = {2 x^{2} - 5 x - 12}_{} g (x) = \frac{4}{2 x + 3} h (x) = 3 x - 12$

Find the sum of two functions that result in a quadratic trinomial. Show your work.

$(f + h) (x) = f (x) + h (x) (f + h) (x) = (2 x^{2} - 5 x - 12) + (3 x - 12) (f + h) (x) = 2 x^{2} - 2 x - 24$

Note

The sum of f and g is a rational function because g is a rational function.

The sum of g and h is a rational function because g is a rational function.

Find the product of two functions that result in a linear binomial. Show your work.

$f (x) \cdot g (x) (2 x^{2} - 5 x - 12) (\frac{4}{2 x + 3}) (2 x + 3) (x - 4) (\frac{4}{2 x + 3}) 4 (x - 4) 4 x - 16$

Note

The product of f and h will be a cubic (degree 3).

The product of g and h will be a rational function.

Evaluate the new functions in parts C and D when $x = 10 .$

$(f + h) (10) = 2 {(10)}^{2} - 2 (10) - 24 (f + h) (10) = 200 - 20 - 24 (f + h) (10) = 156 (f g) (10) = 4 (10) - 16 (f g) (10) = 40 - 16 (f g) (10) = 24$

Note

If you have an error in part C or D, your answers will reflect what is in your functions.

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Evaluate a function.
Determine the sum, difference, product, and/or quotient of functions.
Name the excluded domain values, if any, for combinations of functions.