Study Guide for Algebra: Functions, Transformations, and Operations

Q1. Match the following function with the correct graph: $f(x) = |x|$

Background

Topic: Absolute Value Functions and Graphs

This question tests your ability to recognize the graph of the absolute value function, which is a fundamental parent function in algebra. The absolute value function creates a 'V' shape on the graph, with the vertex at the origin.

Key Terms and Formulas:

• Absolute Value Function: $f(x) = |x|$

• Vertex: The point where the graph changes direction (for $|x|$, this is at $(0,0)$).

• Parent Function: The basic form of a function before any transformations.

Step-by-Step Guidance

1. Recall that the graph of $f(x) = |x|$ is a 'V' shape, opening upwards, with the vertex at the origin.

2. Compare the given graphs to see which one matches this description. Look for a graph with a sharp point at $(0,0)$ and two straight lines extending away from the vertex.

3. Check if any transformations (shifts, stretches, reflections) are present. For the parent function $|x|$, there should be no shifts or stretches.

Try solving on your own before revealing the answer!

Q2. True/False: All vertical transformations affect the x-coordinates of the basic (parent) function.

Background

Topic: Function Transformations

This question tests your understanding of how vertical transformations (such as shifts and stretches) impact the graph of a function, specifically whether they affect the x-coordinates or y-coordinates.

Key Terms:

• Vertical Transformation: Changes to a function that move or stretch the graph up or down.

• Parent Function: The original, untransformed function.

Step-by-Step Guidance

1. Recall that vertical transformations include vertical shifts (adding/subtracting a constant) and vertical stretches/compressions (multiplying by a constant).

2. Think about whether these transformations change the x-values or y-values of points on the graph.

3. Consider an example: $f(x) + c$ (vertical shift) or $a f(x)$ (vertical stretch/compression).

Try solving on your own before revealing the answer!

Q3. Identify which transformation is being involved in the following function: $f(x) = \sqrt{x+7}$

Background

Topic: Function Transformations (Radical Functions)

This question tests your ability to identify the type of transformation applied to a parent function, specifically the square root function.

Key Terms and Formulas:

• Parent Function: $f(x) = \sqrt{x}$

• Transformation: Changes to the function's formula that shift, stretch, or reflect the graph.

• Horizontal Shift: $f(x) = \sqrt{x + c}$ shifts the graph left if $c > 0$, right if $c < 0$.

Step-by-Step Guidance

1. Compare $f(x) = \sqrt{x+7}$ to the parent function $f(x) = \sqrt{x}$.

2. Identify the effect of adding 7 inside the square root. Does it shift the graph left or right?

3. Recall the rule: $f(x) = \sqrt{x+c}$ shifts the graph to the left by $c$ units if $c > 0$.

Try solving on your own before revealing the answer!

Q4. Match the following function with the correct graph: $f(x) = \sqrt{x+6}$

Background

Topic: Square Root Functions and Transformations

This question tests your ability to recognize the graph of a transformed square root function. The parent function $f(x) = \sqrt{x}$ starts at $(0,0)$ and increases slowly. Adding 6 inside the root shifts the graph horizontally.

Key Terms and Formulas:

• Parent Function: $f(x) = \sqrt{x}$

• Horizontal Shift: $f(x) = \sqrt{x+c}$ shifts the graph left by $c$ units if $c > 0$.

Step-by-Step Guidance

1. Identify the parent function $f(x) = \sqrt{x}$ and its graph.

2. Determine how the graph changes when 6 is added inside the square root.

3. Look for a graph that starts at $x = -6$ (since the domain is $x \geq -6$) and follows the shape of the square root function.

Try solving on your own before revealing the answer!

Q5. Graph the following transformed function and its parent function: $f(x) = |x-5|$

Background

Topic: Absolute Value Function Transformations

This question tests your ability to graph both the parent function $f(x) = |x|$ and its horizontally shifted version $f(x) = |x-5|$.

Key Terms and Formulas:

• Parent Function: $f(x) = |x|$

• Horizontal Shift: $f(x) = |x-h|$ shifts the graph right by $h$ units.

Step-by-Step Guidance

1. Graph the parent function $f(x) = |x|$, which is a 'V' shape with vertex at $(0,0)$.

2. For $f(x) = |x-5|$, shift the vertex of the 'V' to $(5,0)$.

3. Draw the two straight lines extending from the new vertex, maintaining the same slope as the parent function.

Try solving on your own before revealing the answer!

Q6. Graph the following transformed function and its parent function: $f(x) = -3x^3$

Background

Topic: Cubic Function Transformations

This question tests your ability to graph the parent cubic function $f(x) = x^3$ and its transformed version $f(x) = -3x^3$, which involves a vertical stretch and a reflection.

Key Terms and Formulas:

• Parent Function: $f(x) = x^3$

• Vertical Stretch: Multiplying by 3 stretches the graph vertically.

• Reflection: Multiplying by -1 reflects the graph across the x-axis.

Step-by-Step Guidance

1. Graph the parent function $f(x) = x^3$, which passes through the origin and has an S-shaped curve.

2. Apply the vertical stretch by multiplying the y-values by 3.

3. Reflect the graph across the x-axis by multiplying the y-values by -1.

Try solving on your own before revealing the answer!

Q7. Let $f(x) = x^2 + 11$ and $g(x) = x + 4$. Find $(f-g)(x)$

Background

Topic: Function Operations (Subtraction)

This question tests your ability to subtract two functions and simplify the result.

Key Terms and Formulas:

• Function Subtraction: $(f-g)(x) = f(x) - g(x)$

Step-by-Step Guidance

1. Write out $f(x)$ and $g(x)$ explicitly: $f(x) = x^2 + 11$, $g(x) = x + 4$.

2. Subtract $g(x)$ from $f(x)$: $(f-g)(x) = (x^2 + 11) - (x + 4)$.

3. Simplify the expression by combining like terms.

Try solving on your own before revealing the answer!

Q8. Let $f(x) = x^2 + 11$ and $g(x) = x + 4$. Find $(f+g)(3)$

Background

Topic: Function Operations (Addition and Evaluation)

This question tests your ability to add two functions and evaluate the result at a specific value.

Key Terms and Formulas:

• Function Addition: $(f+g)(x) = f(x) + g(x)$

• Evaluation: Substitute $x = 3$ into the resulting expression.

Step-by-Step Guidance

1. Write out $f(x)$ and $g(x)$: $f(x) = x^2 + 11$, $g(x) = x + 4$.

2. Add the functions: $(f+g)(x) = (x^2 + 11) + (x + 4)$.

3. Substitute $x = 3$ into the expression and simplify.

Try solving on your own before revealing the answer!

Q9. Let $g(x) = x + 4$. Find $(g \cdot g)(x)$

Background

Topic: Function Multiplication

This question tests your ability to multiply a function by itself.

Key Terms and Formulas:

• Function Multiplication: $(g \cdot g)(x) = g(x) \cdot g(x)$

Step-by-Step Guidance

1. Write out $g(x)$: $g(x) = x + 4$.

2. Multiply $g(x)$ by itself: $(g \cdot g)(x) = (x + 4) \cdot (x + 4)$.

3. Expand the expression using the distributive property.

Try solving on your own before revealing the answer!

Q10. Let $f(x) = x^2 + 11$ and $g(x) = x + 4$. Find $(g \circ f)(x)$

Background

Topic: Function Composition

This question tests your ability to compose two functions, meaning you substitute one function into another.

Key Terms and Formulas:

• Function Composition: $(g \circ f)(x) = g(f(x))$

Step-by-Step Guidance

1. Write out $f(x)$ and $g(x)$: $f(x) = x^2 + 11$, $g(x) = x + 4$.

2. Substitute $f(x)$ into $g(x)$: $g(f(x)) = f(x) + 4$.

3. Write the resulting expression in terms of $x$.

Try solving on your own before revealing the answer!

Q11. Let $f(x) = \sqrt{1-x}$ and $h(x) = 4x^3$. Find $(f-h)(-3)$

Background

Topic: Function Operations (Subtraction and Evaluation)

This question tests your ability to subtract two functions and evaluate the result at a specific value.

Key Terms and Formulas:

• Function Subtraction: $(f-h)(x) = f(x) - h(x)$

• Evaluation: Substitute $x = -3$ into the resulting expression.

Step-by-Step Guidance

1. Write out $f(x)$ and $h(x)$: $f(x) = \sqrt{1-x}$, $h(x) = 4x^3$.

2. Subtract $h(x)$ from $f(x)$: $(f-h)(x) = \sqrt{1-x} - 4x^3$.

3. Substitute $x = -3$ into the expression and simplify.

Try solving on your own before revealing the answer!

Q12. Let $f(x) = \sqrt{1-x}$ and $h(x) = 4x^3$. Find $(f \cdot h)(3)$

Background

Topic: Function Operations (Multiplication and Evaluation)

This question tests your ability to multiply two functions and evaluate the result at a specific value.

Key Terms and Formulas:

• Function Multiplication: $(f \cdot h)(x) = f(x) \cdot h(x)$

• Evaluation: Substitute $x = 3$ into the resulting expression.

Step-by-Step Guidance

1. Write out $f(x)$ and $h(x)$: $f(x) = \sqrt{1-x}$, $h(x) = 4x^3$.

2. Multiply the functions: $(f \cdot h)(x) = \sqrt{1-x} \cdot 4x^3$.

3. Substitute $x = 3$ into the expression and simplify.

Try solving on your own before revealing the answer!

Q13. Let $f(x) = \sqrt{1-x}$ and $h(x) = 4x^3$. Find $(h \cdot f)(x)$

Background

Topic: Function Operations (Multiplication)

This question tests your ability to multiply two functions and simplify the result.

Key Terms and Formulas:

• Function Multiplication: $(h \cdot f)(x) = h(x) \cdot f(x)$

Step-by-Step Guidance

1. Write out $h(x)$ and $f(x)$: $h(x) = 4x^3$, $f(x) = \sqrt{1-x}$.

2. Multiply the functions: $(h \cdot f)(x) = 4x^3 \cdot \sqrt{1-x}$.

3. Simplify the expression as much as possible.

Try solving on your own before revealing the answer!

Q14. Let $h(x) = 4x^3$. Find $(h \circ h)(x)$

Background

Topic: Function Composition

This question tests your ability to compose a function with itself.

Key Terms and Formulas:

• Function Composition: $(h \circ h)(x) = h(h(x))$

Step-by-Step Guidance

1. Write out $h(x)$: $h(x) = 4x^3$.

2. Substitute $h(x)$ into itself: $h(h(x)) = 4(h(x))^3$.

3. Expand the expression to write it in terms of $x$.

Try solving on your own before revealing the answer!

Q15. For each equation, determine whether it is "odd", "even", or neither: $g(x) = 4x^3 - x$

Background

Topic: Function Symmetry (Odd/Even Functions)

This question tests your ability to classify functions as odd, even, or neither based on their symmetry properties.

Key Terms and Formulas:

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain.

Step-by-Step Guidance

1. Substitute $-x$ into the function: $g(-x) = 4(-x)^3 - (-x)$.

2. Simplify the expression and compare it to $g(x)$ and $-g(x)$.

3. Determine if the function is odd, even, or neither based on the result.

Try solving on your own before revealing the answer!

Q16. For each equation, determine whether it is "odd", "even", or neither: $f(x) = 2x^2 - 5$

Background

Topic: Function Symmetry (Odd/Even Functions)

This question tests your ability to classify quadratic functions as odd, even, or neither.

Key Terms and Formulas:

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain.

Step-by-Step Guidance

1. Substitute $-x$ into the function: $f(-x) = 2(-x)^2 - 5$.

2. Simplify the expression and compare it to $f(x)$ and $-f(x)$.

3. Determine if the function is odd, even, or neither based on the result.

Try solving on your own before revealing the answer!

Q17. For each equation, determine whether it is "odd", "even", or neither: $h(x) = 2x^2 - 3x + 7$

Background

Topic: Function Symmetry (Odd/Even Functions)

This question tests your ability to classify polynomial functions as odd, even, or neither.

Key Terms and Formulas:

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain.

Step-by-Step Guidance

1. Substitute $-x$ into the function: $h(-x) = 2(-x)^2 - 3(-x) + 7$.

2. Simplify the expression and compare it to $h(x)$ and $-h(x)$.

3. Determine if the function is odd, even, or neither based on the result.

Try solving on your own before revealing the answer!