Question

an equation that defines y as a function of x is given. a. solve for y in terms of x and replace y with the function notation f(x). b. find f(3).
y + 3x² = 5 - 5x
a. solve for y in terms of x and replace y with the function notation f(x).
f(x) =
b. find f(3).
f(3) =

Explanation:

Part a

Step 1: Isolate \( y \)

We start with the equation \( y + 3x^2 = 5 - 0.5x \). To solve for \( y \), we subtract \( 3x^2 \) from both sides of the equation.
\[
y + 3x^2 - 3x^2 = 5 - 0.5x - 3x^2
\]

Step 2: Simplify

Simplifying the left side (since \( 3x^2 - 3x^2 = 0 \)) and writing the right side in standard form, we get:
\[
y = -3x^2 - 0.5x + 5
\]
Now, we replace \( y \) with \( f(x) \), so \( f(x)= -3x^2 - 0.5x + 5 \).

Part b

Step 1: Substitute \( x = 3 \) into \( f(x) \)

We have \( f(x)= -3x^2 - 0.5x + 5 \). To find \( f(3) \), we substitute \( x = 3 \) into the function:
\[
f(3)= -3(3)^2 - 0.5(3) + 5
\]

Step 2: Calculate the powers and multiplications

First, calculate \( (3)^2 = 9 \), then \( -3\times9=-27 \), and \( -0.5\times3 = -1.5 \):
\[
f(3)= -27 - 1.5 + 5
\]

Step 3: Perform the additions and subtractions

First, \( -27 - 1.5=-28.5 \), then \( -28.5 + 5=-23.5 \).

Answer:

s:
a. \( f(x)=\boldsymbol{-3x^2 - 0.5x + 5} \)

b. \( f(3)=\boldsymbol{-23.5} \) (or \( -\frac{47}{2} \) if we prefer a fraction)