Functions And Relations (20-1)

Question 1

Determine if f(x)=x²+3x+2 is a function.

Answer 1

Yes, it passes the vertical line test.

Question 2

State the domain of f(x)=x²+3x+2.

Answer 2

Domain: all real numbers

Question 3

State the range of f(x)=x²+3x+2.

Answer 3

Range: y ≥ -0.25

(vertex y-value)

Complete the square:
f(x)=x²+3x+2

Think divide by 2, then square

f(x)=(x²+3x + 1.5² - 1.5²) +2
f(x)=(x²+3x + 2.25) + 2 - 2.25
f(x)=(x²+3x + 2.25) - 0.25
f(x)=(x+1.5)² - 0.25

Vertex = (-1.5, -0.25)
Parabola end behavior is quadrant 2 to quadrant 1 (smile), so we have a minimum.

Range
Interval notation: [-0.25, infinity)
Set notation: {y|yER, y >= -0.25}

Graphing calculator:
Enter the function in Y1: .

Press GRAPH to see the parabola.

Press 2nd → TRACE → 3: minimum.

Move the cursor near the bottom of the parabola. The calculator displays the vertex coordinates automatically.

Tip: You can also check the value by using the TABLE feature near X = -1.5 to see that .

Question 4

Find f(4) for f(x)=x²-5x+6.

Answer 4

f(x)=x² - 5x + 6
f(4)= 4² - 5(4) + 6
f(4)= 16 - 20 + 6
f(4) = 2

Question 5

Apply transformation: y=(x-3)²+5 — what is the vertex?

Answer 5

Vertex: (3, 5)

Question 6

Apply transformation: y=-2|x+4| — state reflection.

Answer 6

Reflection over x-axis

1. Start with the parent function: (V-shaped graph, vertex at (0,0)).
2. Transformation steps:

Horizontal shift: → shifts the graph 4 units left.

Vertical stretch: in front → stretch vertically by a factor of 2.

Reflection: The negative sign reflects the graph over the x-axis, so the “V” opens downward.

3. Vertex: After shift and reflection, vertex is at (-4,0).
4. Graphing calculator usage:

Enter in Y1.

Press GRAPH to visualize.

Use 2nd → TRACE → maximum to find the vertex if needed (since it’s reflected, the vertex is now the maximum).

Summary: Reflection is over the x-axis.

Question 7

Determine if relation {(1,3),(2,5),(3,5)} is a function.

Answer 7

Yes, each x has exactly one y.

Question 8

Determine if relation {(1,3),(1,4)} is a function.

Answer 8

No, x=1 maps to two different y-values.

Question 9

Find (f∘g)(x) for f(x)=x+1, g(x)=2x.

Answer 9

(f∘g)(x)=2x+1

Question 10

Find inverse of f(x)=2x-6.

Answer 10

f⁻¹(x)=(x+6)/2

Question 11

Solve for x if f(x)=x²+2x, f(x)=0.

Answer 11

x=0 or x=-2

Question 12

Identify vertical stretch factor in y=3(x-2)².

Answer 12

Vertical stretch factor = 3