Chapter 5 Worksheet: Inverse and Logarithmic Functions

Vocabulary and key evaluation terms covering inverse functions and basic logarithmic simplifications from Chapter 5 (Sections 5.1 and 5.2).

$h^{-1}(x)$ for $h(x) = \sqrt{5-x}$

The inverse function formula found by swapping variables and solving for $y$, yielding $h^{-1}(x) = 5 - x^2$.

$$h^{-1}(3)$$

The value $-4$, found either graphically from $h(x) = \sqrt{5-x}$ or algebraically by evaluating $5 - (3)^2$.

$f^{-1}(5)$ for $f(t) = \frac{2}{3}t + 1$

The value $6$, obtained by finding the inverse function $f^{-1}(x) = \frac{3}{2}(x - 1)$ and evaluating it at $x = 5$.

$m^{-1}(-3)$ for $m(v) = 6 - \frac{2}{v}$

The value $\frac{2}{9}$, determined by deriving the inverse formula $m^{-1}(x) = \frac{-2}{x - 6}$ and evaluating it at $x = -3$.

$f(a) + f(b)$ versus $f(a+b)$ for $f(x) = \log(x)$

Two expressions that are not equal; for example, $f(487) + f(206) \approx 5.001$, whereas $f(487 + 206) \approx 2.841$.

$$10^{\log(2k)}$$

An exponential-logarithmic expression that simplifies directly to $2k$.

$$10^{3\log(x)}$$

An expression that simplifies to $x^3$ by applying the power property of logarithms to obtain $10^{\log(x^3)}$.

$$\log(10^{1-x})$$

A base-10 logarithmic expression that simplifies directly to $1-x$.