Precalculus Test 4 Study Guide: Inequalities, Functions, Logarithms, and Exponentials

Q1(a). Solve the inequality: $x + 12 < 7$

Background

Topic: Linear Inequalities

This question tests your ability to solve a basic linear inequality and express the solution in interval notation.

Key Terms and Formulas:

• Linear inequality: An inequality involving a linear expression.

• Interval notation: A way to describe sets of numbers between endpoints.

Step-by-Step Guidance

1. Start by isolating $x$ on one side of the inequality. Subtract $12$ from both sides:

   $x + 12 - 12 < 7 - 12$

2. Simplify both sides to get an inequality in terms of $x$.

Try solving on your own before revealing the answer!

Q1(b). Solve the inequality: $2x^3 > -8x^2$

Background

Topic: Polynomial Inequalities

This question tests your ability to solve inequalities involving polynomials by factoring and analyzing sign changes.

Key Terms and Formulas:

• Factoring: Expressing a polynomial as a product of its factors.

• Test intervals: Checking the sign of the expression in different intervals determined by the roots.

Step-by-Step Guidance

1. Move all terms to one side to set the inequality to zero:

   $2x^3 + 8x^2 > 0$

2. Factor out the greatest common factor from the expression.

3. Set each factor equal to zero to find the critical points.

4. Use a sign chart or test values in each interval to determine where the expression is positive.

Try solving on your own before revealing the answer!

Q1(c). Solve the inequality: $\frac{x-1}{x+2} \geq -2$

Background

Topic: Rational Inequalities

This question tests your ability to solve inequalities involving rational expressions and to express the solution in interval notation.

Key Terms and Formulas:

• Rational expression: A fraction with polynomials in the numerator and denominator.

• Critical points: Values where the numerator or denominator is zero.

Step-by-Step Guidance

1. Bring all terms to one side to set the inequality to zero:

   $\frac{x-1}{x+2} + 2 \geq 0$

2. Combine the terms over a common denominator.

3. Simplify the numerator and set the expression $\geq 0$.

4. Find the critical points by setting the numerator and denominator equal to zero.

5. Test intervals between the critical points to determine where the expression is non-negative.

Try solving on your own before revealing the answer!

Q1(d). Solve the inequality: $\frac{x-1}{x} \geq 0$

Background

Topic: Rational Inequalities

This question tests your ability to solve inequalities involving rational expressions and to express the solution in interval notation.

Key Terms and Formulas:

• Rational expression: A fraction with polynomials in the numerator and denominator.

• Critical points: Values where the numerator or denominator is zero.

Step-by-Step Guidance

1. Identify the values where the numerator and denominator are zero:

   Set $x-1 = 0$ and $x = 0$.

2. Determine the sign of the expression in each interval defined by these critical points.

3. Include points where the expression equals zero, if allowed by the inequality.

4. Express the solution in interval notation, being careful with open and closed intervals.

Try solving on your own before revealing the answer!

Q2. Find the domain of $f(x) = \ln\left(\frac{x-1}{x+1}\right)$, expressing your answer in interval notation.

Background

Topic: Domain of Logarithmic Functions

This question tests your understanding of the domain restrictions for logarithmic functions, especially when the argument is a rational expression.

Key Terms and Formulas:

• Domain: The set of all real numbers for which the function is defined.

• Logarithm: $\ln(x)$ is defined only for $x > 0$.

Step-by-Step Guidance

1. Set the argument of the logarithm greater than zero:

   $\frac{x-1}{x+1} > 0$

2. Solve the inequality by finding where the numerator and denominator have the same sign.

3. Express the solution in interval notation, excluding any points where the denominator is zero.

Try solving on your own before revealing the answer!

Q2(a). For $f(x) = \frac{x}{x+3}$ and $g(x) = \frac{2}{x}$, find the domains of $f \circ g$ and $g \circ f$. Then calculate $(f \circ g)(x)$ and $(g \circ f)(x)$.

Background

Topic: Composition of Functions and Domains

This question tests your ability to find the domain of composite functions and to compute the composition explicitly.

Key Terms and Formulas:

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

• Domain: The set of all $x$ for which the composition is defined.

Step-by-Step Guidance

1. Find the domain of $g(x)$ by identifying values that make the denominator zero.

2. Find the domain of $f(x)$ by identifying values that make its denominator zero.

3. For $f \circ g$, determine all $x$ such that $g(x)$ is defined and $f(g(x))$ is defined.

4. For $g \circ f$, determine all $x$ such that $f(x)$ is defined and $g(f(x))$ is defined.

5. Compute $(f \circ g)(x)$ by substituting $g(x)$ into $f(x)$, and $(g \circ f)(x)$ by substituting $f(x)$ into $g(x)$.

Try solving on your own before revealing the answer!

Q2(b). For $f(x) = 3x^2 - 7$, $g(x) = 2x - a$, find $a$ such that the graph of $f \circ g$ crosses the $y$-axis at $68$.

Background

Topic: Function Composition and Intercepts

This question tests your understanding of function composition and how to find parameters so that the composite function passes through a specific point.

Key Terms and Formulas:

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

• $y$-intercept: The value of the function when $x = 0$.

Step-by-Step Guidance

1. Write the composition $f(g(x))$ explicitly.

2. Set $x = 0$ to find the $y$-intercept of $f \circ g$.

3. Set $f(g(0)) = 68$ and solve for $a$.

Try solving on your own before revealing the answer!

Q2(c). For $f(x) = x + \frac{1}{x}$, $g(x) = 1 - x$, compute $(f \circ g)(0)$ and $(g \circ f)(1)$.

Background

Topic: Function Composition and Evaluation

This question tests your ability to evaluate composite functions at specific points.

Key Terms and Formulas:

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

• Evaluation: Substitute the given value into the function.

Step-by-Step Guidance

1. Compute $g(0)$, then substitute this value into $f(x)$ to find $f(g(0))$.

2. Compute $f(1)$, then substitute this value into $g(x)$ to find $g(f(1))$.

Try solving on your own before revealing the answer!