Precalculus Study Guide: Functions, Graphs, Exponentials, Logarithms, and Geometry

Q1. State whether the following statement is true or false: If $f(x) = x^2$ and $g(x) = \sqrt{x+9}$, then $(f \circ g)(4) = 5$.

Background

Topic: Function Composition

This question tests your understanding of how to compose two functions and evaluate the result at a specific input.

Key Terms and Formulas:

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

• $f(x) = x^2$

• $g(x) = \sqrt{x+9}$

Step-by-Step Guidance

1. Start by evaluating $g(4)$: $g(4) = \sqrt{4+9}$.

2. Calculate the value inside the square root and simplify $g(4)$.

3. Next, use the result from $g(4)$ as the input for $f(x)$: $f(g(4))$.

4. Set up $f(g(4))$ using the value you found for $g(4)$.

Try solving on your own before revealing the answer!

Final Answer: False

$(f \circ g)(4) = f(g(4)) = f(\sqrt{13}) = (\sqrt{13})^2 = 13$, not 5.

The statement is false because the composition yields 13, not 5.

Q2. Change the exponential equation $e^x = 8$ to logarithmic form.

Background

Topic: Exponential and Logarithmic Equations

This question tests your ability to convert between exponential and logarithmic forms.

Key Terms and Formulas:

• Exponential form: $a^x = b$

• Logarithmic form: $x = \log_a b$

• Natural logarithm: $\ln$ is logarithm base $e$

Step-by-Step Guidance

1. Identify the base of the exponential equation ($e$ in this case).

2. Recall the conversion: $a^x = b$ is equivalent to $x = \log_a b$.

3. Apply this conversion to $e^x = 8$.

Try solving on your own before revealing the answer!

Final Answer: $x = \ln 8$

The logarithmic form of $e^x = 8$ is $x = \ln 8$.

This uses the natural logarithm because the base is $e$.

Q3. Solve the equation $2x^2 = 16$. (Type an integer or a fraction. Use a comma to separate answers as needed.)

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations by isolating the variable and taking square roots.

Key Terms and Formulas:

• Quadratic equation: $ax^2 + bx + c = 0$

• Square root property: If $x^2 = a$, then $x = \pm \sqrt{a}$

Step-by-Step Guidance

1. Divide both sides of the equation by 2 to isolate $x^2$.

2. Take the square root of both sides, remembering to include both positive and negative roots.

Try solving on your own before revealing the answer!

Final Answer: $x = 2, x = -2$

Dividing by 2 gives $x^2 = 8$, so $x = \pm \sqrt{8} = \pm 2\sqrt{2}$.

Both positive and negative values are solutions.

Q4. Write the standard form of the equation and the general form of the circle of radius $r = 5$ and center $(h, k) = (0, 0)$. Graph the circle.

Background

Topic: Circles in Coordinate Geometry

This question tests your understanding of the equation of a circle and how to graph it.

Key Terms and Formulas:

• Standard form: $(x-h)^2 + (y-k)^2 = r^2$

• General form: $x^2 + y^2 + Dx + Ey + F = 0$

Step-by-Step Guidance

1. Plug the center $(h, k) = (0, 0)$ and radius $r = 5$ into the standard form equation.

2. Expand the equation if needed to get the general form.

3. Sketch or visualize the circle on a coordinate plane with center at the origin and radius 5.

Try solving on your own before revealing the answer!

Final Answer: Standard form: $x^2 + y^2 = 25$

The general form is $x^2 + y^2 - 25 = 0$.

The circle is centered at the origin with radius 5.

Q5. Find the distance $d(P_1, P_2)$ between the points $P_1 = (0, 0)$ and $P_2 = (1, 4)$. (Simplify your answer. Type an exact answer, using radicals as needed.)

Background

Topic: Distance Formula in Coordinate Geometry

This question tests your ability to use the distance formula to find the length between two points in the plane.

Key Terms and Formulas:

• Distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step-by-Step Guidance

1. Identify the coordinates: $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$.

2. Plug the values into the distance formula.

3. Simplify the expression under the square root.

Try solving on your own before revealing the answer!

Final Answer: $d = \sqrt{17}$

Plugging in the values gives $d = \sqrt{(1-0)^2 + (4-0)^2} = \sqrt{1 + 16} = \sqrt{17}$.

This is the exact distance between the two points.