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 paired with exactly 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 ordered pairs, the inverse is found by swapping each pair: $(a, b) \rightarrow (b, a)$.

• For an algebraic function $y = f(x)$, solve for $x$ in terms of $y$, then switch $x$ and $y$.

Step-by-Step Guidance

1. For the set of ordered pairs, swap the $x$ and $y$ values in each pair.

2. For the algebraic function $y = \frac{5x-3}{2}$:

   1. Multiply both sides by 2 to clear the denominator.

   2. Add 3 to both sides to isolate the $5x$ term.

   3. Divide both sides by 5 to solve for $x$.

   4. Switch $x$ and $y$ to write the inverse function.

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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. For quarterly or monthly compounding, use $A = P \left(1 + \frac{r}{n}\right)^{nt}$.

3. For continuous compounding, use $A = Pe^{rt}$.

4. Plug the values into the appropriate formula, being careful to convert the interest rate to a decimal (e.g., 8% becomes 0.08).

5. Calculate the exponent $nt$ or $rt$ as needed.

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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 basic logarithm properties and your ability to simplify logarithmic expressions.

Key Properties:

• $\log_b b^k = k$

• $\ln e^k = k$

• $\log_b (a^k) = k \log_b a$

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

Step-by-Step Guidance

1. For each logarithm, identify if the argument is a power or root of the base.

2. Apply the appropriate property to simplify the expression.

3. For square roots, rewrite as a fractional exponent before simplifying.

4. For natural logarithms ($\ln$), recall that $\ln e^k = k$.

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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 in base 10 or $e$.

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. Identify the base $b$ and the argument $a$ for each logarithm.

2. Apply the change of base formula: $\log_b a = \frac{\log a}{\log b}$ (using common or natural logarithms).

3. Set up the expression for each logarithm using your calculator's log or ln function.

4. Do not compute the final value yet; just set up the calculation.

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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 terms into a single logarithm using properties of logarithms.

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. Use the power rule to move coefficients in front of logs to exponents inside the log.

2. Use the product rule to combine sums of logs into a single log of a product.

3. Use the quotient rule to combine differences of logs into a single log of a quotient.

4. Simplify the resulting expression as much as possible.

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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.

Background

Topic: Logarithm Properties and Evaluating Logarithms

This question tests your ability to use properties of logarithms (product, quotient, and power rules) to find the logarithm of other numbers based on given values.

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 argument (e.g., 15, $\frac{5}{17}$, 75) in terms of the numbers you have logarithms for (3, 5, 17).

2. Apply the product, quotient, or power rules as needed to write the desired logarithm in terms of the given values.

3. Set up the calculation using the provided logarithm values, but do not compute the final result yet.

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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 rewriting both sides with the same base or by using logarithms.

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. For each equation, try to express both sides with the same base if possible.

2. If not, apply the logarithm to both sides to bring the exponent down.

3. Use properties of logarithms to isolate the variable.

4. Set up the equation for the final calculation, but do not solve for the variable yet.

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

Background

Topic: Logarithmic Equations and Domain Considerations

This question tests your ability to solve equations involving logarithms and to check whether 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 that the solution makes all logarithm arguments positive (no extraneous solutions).

Step-by-Step Guidance

1. Use logarithm properties to combine terms on each side.

2. Rewrite the equation in exponential form.

3. Solve for the variable algebraically.

4. Check the solution in the original equation to ensure all arguments are positive.

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Q10. Exponential Growth: Porsche Value Problem

Background

Topic: Exponential Growth Functions

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 $V$ is the value at time $t$, $V_0$ is the initial value, $k$ is the growth constant, and $t$ is time in years.

• Doubling time: $t_{double} = \frac{\ln 2}{k}$

Step-by-Step Guidance

1. Set up the exponential growth equation using the given values for $V_0$, $V$, and $t$.

2. Solve for $k$ by substituting the known values and isolating $k$ using logarithms.

3. Write the exponential growth function using the value of $k$.

4. For predictions, substitute the desired $t$ value into the function and solve for $V$.

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

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