Question

study guide and intervention (continued)
relations and functions
equations of relations and functions equations that represent functions are often written in functional notation. for example, y = 10 - 8x can be written as f(x)=10 - 8x. this notation emphasizes the fact that the values of y, the dependent variable, depend on the values of x, the independent variable.
to evaluate a function, or find a functional value, means to substitute a given value in the domain into the equation to find the corresponding element in the range.
example: given f(x)=x² + 2x, find each value.
a. f(3)
f(x)=x² + 2x
f(3)=3² + 2(3)
=15
b. f(5a)
f(x)=x² + 2x
f(5a)=(5a)² + 2(5a)
=25a² + 10a
exercises
graph each relation or equation and determine the domain and range. determine whether the relation is a function, is one - to - one, onto, both, or neither. then state whether it is discrete or continuous.

1. y = 3
2. y = x² - 1
3. y = 3x + 2

find each value if f(x)=-2x + 4.

1. f(12)
2. f(6)
3. f(2b)

find each value if g(x)=x³ - x.

1. g(5)
2. g(-2)
3. g(7c)

Explanation:

Step1: Evaluate \(f(12)\) for \(f(x)=-2x + 4\)

Substitute \(x = 12\) into \(f(x)\):
\[

$$\begin{align*} f(12)&=-2\times12 + 4\\ &=-24+4\\ &=- 20 \end{align*}$$

\]

Step2: Evaluate \(f(6)\) for \(f(x)=-2x + 4\)

Substitute \(x = 6\) into \(f(x)\):
\[

$$\begin{align*} f(6)&=-2\times6+4\\ &=-12 + 4\\ &=-8 \end{align*}$$

\]

Step3: Evaluate \(f(2b)\) for \(f(x)=-2x + 4\)

Substitute \(x = 2b\) into \(f(x)\):
\[

$$\begin{align*} f(2b)&=-2\times(2b)+4\\ &=-4b + 4 \end{align*}$$

\]

Step4: Evaluate \(g(5)\) for \(g(x)=x^{3}-x\)

Substitute \(x = 5\) into \(g(x)\):
\[

$$\begin{align*} g(5)&=5^{3}-5\\ &=125-5\\ &=120 \end{align*}$$

\]

Step5: Evaluate \(g(-2)\) for \(g(x)=x^{3}-x\)

Substitute \(x=-2\) into \(g(x)\):
\[

$$\begin{align*} g(-2)&=(-2)^{3}-(-2)\\ &=-8 + 2\\ &=-6 \end{align*}$$

\]

Step6: Evaluate \(g(7c)\) for \(g(x)=x^{3}-x\)

Substitute \(x = 7c\) into \(g(x)\):
\[

$$\begin{align*} g(7c)&=(7c)^{3}-7c\\ &=343c^{3}-7c \end{align*}$$

\]

Answer:

1. \(f(12)=-20\)
2. \(f(6)=-8\)
3. \(f(2b)=-4b + 4\)
4. \(g(5)=120\)
5. \(g(-2)=-6\)
6. \(g(7c)=343c^{3}-7c\)