College Algebra Study Guide: Step-by-Step Guidance

Q1. Are the following functions one-to-one?

Background

Topic: Functions and One-to-One (Injective) Functions

This question tests your understanding of what it means for a function to be one-to-one (injective). A function is one-to-one if every output value is produced by only one input value (no two different inputs have the same output).

Key Terms:

• Function: A relation where each input has exactly one output.

• One-to-one (Injective): A function where each output is paired with only one input.

Step-by-Step Guidance

1. For each set of ordered pairs, look for repeated output values (the second number in each pair).

2. If any output value is repeated for different input values, the function is not one-to-one.

3. If all output values are unique, the function is one-to-one.

4. Go through each set and check for repeated outputs.

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Q2. Find the inverse of the following functions.

Background

Topic: Inverse Functions

This question tests your ability to find the inverse of a function, both for a set of ordered pairs and for an algebraic function.

Key Terms and Formulas:

• Inverse Function: A function that "undoes" the action of the original function. If $f(x)$ maps $a$ to $b$, then $f^{-1}(x)$ maps $b$ back to $a$.

• For a set of points: Swap each $x$ and $y$ value in the ordered pairs.

• For an equation: Solve for $x$ in terms of $y$, then switch $x$ and $y$.

Step-by-Step Guidance

1. For the set of points, write each pair as $(y, x)$ instead of $(x, y)$.

2. For the equation $y = \frac{5x-3}{2}$, swap $x$ and $y$ to get $x = \frac{5y-3}{2}$.

3. Solve this new equation for $y$.

4. Express the inverse function as $f^{-1}(x)$.

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Q3. Future Value with Compound Interest

Background

Topic: Exponential Functions and Compound Interest

This question tests your ability to use compound interest formulas to find the future value of an investment.

Key Formulas:

• For compounding $n$ times per year: $A = P \left(1 + \frac{r}{n}\right)^{nt}$

• For continuous compounding: $A = Pe^{rt}$

• Where:

  • $A$ = future value

  • $P$ = initial deposit (principal)

  • $r$ = annual interest rate (as a decimal)

  • $n$ = number of compounding periods per year

  • $t$ = number of years

Step-by-Step Guidance

1. Identify the values for $P$, $r$, $n$, and $t$ from the problem statement.

2. Convert the interest rate from a percent to a decimal by dividing by 100.

3. Plug the values into the appropriate formula (quarterly, monthly, or continuous compounding).

4. Calculate the exponent $nt$ (for non-continuous) or $rt$ (for continuous).

5. Set up the expression for $A$ but do not compute the final value yet.

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Q4. Completely simplify the following logarithms without using the change of base formula.

Background

Topic: Logarithm Properties and Simplification

This question tests your understanding of logarithm properties, such as the power rule, product rule, and the definition of logarithms.

Key Properties:

• $\log_b b^k = k$

• $\log_b \sqrt{a} = \frac{1}{2} \log_b a$

• $\ln e^k = k$

Step-by-Step Guidance

1. Rewrite each logarithm using exponent or root properties if possible.

2. Apply the appropriate logarithm property (such as the power rule or the definition of logarithms).

3. Simplify the expression as much as possible, but stop before the final numeric value.

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Q5. Find the following logarithms by using the change of base formula.

Background

Topic: Logarithms and Change of Base Formula

This question tests your ability to use the change of base formula to evaluate logarithms that are not base 10 or base $e$.

Key Formula:

• $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive base (commonly 10 or $e$).

Step-by-Step Guidance

1. Write the change of base formula for each logarithm.

2. Choose a base for your calculation (usually 10 or $e$).

3. Set up the expression for each logarithm using your chosen base, but do not compute the final value.

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Q6. Express the following expressions as a single logarithm and completely simplify.

Background

Topic: Logarithm Properties (Product, Quotient, and Power Rules)

This question tests your ability to combine multiple logarithmic expressions into a single logarithm using logarithm properties.

Key Properties:

• $a \log_b x = \log_b x^a$ (Power Rule)

• $\log_b x + \log_b y = \log_b (xy)$ (Product Rule)

• $\log_b x - \log_b y = \log_b \left(\frac{x}{y}\right)$ (Quotient Rule)

Step-by-Step Guidance

1. Apply the power rule to coefficients in front of logarithms.

2. Use the product rule to combine sums of logarithms.

3. Use the quotient rule to combine differences of logarithms.

4. Simplify the resulting expression as much as possible, but stop before the final simplification.

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Q7. If $\log_b 3 = 0.415$, $\log_b 5 = 0.568$, and $\log_b 17 = 1.873$, find the following logarithms rounded to the nearest thousandth.

Background

Topic: Logarithm Properties and Evaluations

This question tests your ability to use given logarithm values and properties to find the logarithm of other numbers.

Key Properties:

• $\log_b (xy) = \log_b x + \log_b y$

• $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$

• $\log_b (x^k) = k \log_b x$

Step-by-Step Guidance

1. Express each number (like 15, $\frac{5}{17}$, 75) in terms of 3, 5, and 17.

2. Apply the appropriate logarithm property to break down the expression.

3. Substitute the given logarithm values into your expression.

4. Set up the calculation, but do not compute the final value.

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Q8. Solve the following exponential equations.

Background

Topic: Exponential Equations

This question tests your ability to solve equations where the variable is in the exponent, often by using logarithms or by rewriting both sides with the same base.

Key Steps:

• Rewrite both sides of the equation with the same base if possible.

• If not possible, take the logarithm of both sides to bring the exponent down.

• Solve for the variable using algebraic manipulation.

Step-by-Step Guidance

1. Try to express both sides of the equation with the same base, or take the logarithm of both sides.

2. Use logarithm properties to bring the exponent down.

3. Solve for the variable step by step, but stop before the final calculation.

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Q9. Solve the following logarithmic equations. State whether each solution is valid or extraneous.

Background

Topic: Logarithmic Equations and Domain

This question tests your ability to solve equations involving logarithms and to check if the solutions are valid (i.e., the arguments of all logarithms are positive).

Key Steps:

• Combine logarithms using properties to get a single logarithm on each side if possible.

• Rewrite the equation in exponential form to solve for the variable.

• Check the solution by substituting back into the original equation to ensure all logarithms are defined (arguments are positive).

Step-by-Step Guidance

1. Combine the logarithms on each side using the product or quotient rule.

2. Rewrite the equation in exponential form.

3. Solve for the variable, but stop before the final calculation.

4. Remember to check if your solution makes all logarithm arguments positive.

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Q10. The market value of the 1993-1995 Porsche 928 has had a recent increase in value from $8000 in 2000 to $15400 in 2015. Assuming that the value,$V$ of the car has grown exponentially:

Background

Topic: Exponential Growth Models

This question tests your ability to model real-world exponential growth, find the growth constant, write the exponential function, and use it to make predictions.

Key Formula:

• $V = V_0 e^{kt}$, where $V_0$ is the initial value, $k$ is the growth constant, and $t$ is time in years.

• To find $k$, use two data points and solve for $k$.

• To find doubling time, use $t = \frac{\ln 2}{k}$.

Step-by-Step Guidance

1. Write the exponential growth equation using the given initial value.

2. Plug in the second data point to solve for $k$.

3. Once you have $k$, write the full exponential model.

4. Use the model to estimate the value at a given year, or to solve for the time when the value reaches a certain amount.

5. For doubling time, use the formula $t = \frac{\ln 2}{k}$.

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