When graphing without a calculator, we use the fact that the inverse of a logarithmic function is an exponential function. More links and references related to the inverse functions. The radioactive bismuth isotope $^{210} \mathrm{Bi}$ has a half-life of 5 days. Some lending institutions calculate the monthly payment $M$ on a loan of $L$ dollars at an interest rate $r$ (expressed as a decimal) by using the formula $$M=\frac{L r k}{12(k-1)}$$ where $k=[1+(r / 12)]^{12 t}$ and $t$ is the number of years that the loan is in effect.Find the largest 25 -year home mortgage that can be obtained at an interest rate of $7 \%$ if the monthly payment is to be 1500 dollars. (b) If $N_{0}=200,$ sketch the graph of $N$ for $0 \leq t \leq 5$. Sketch the graph of $f$ if $a=2$. (b) At the end of $T$ years, the item has a salvage value of $s$ dollars. Some lending institutions calculate the monthly payment $M$ on a loan of $L$ dollars at an interest rate $r$ (expressed as a decimal) by using the formula $$M=\frac{L r k}{12(k-1)}$$ where $k=[1+(r / 12)]^{12 t}$ and $t$ is the number of years that the loan is in effect.Business loan The owner of a small business decides to finance a new computer by borrowing 3000 dollars for 2 years at an interest rate of $7.5 \%$(a) Find the monthly payment. Solutions ... Graph. New content will be added above the current area of focus upon selection This reflects the graph about the line y=x. (a) $f(x)=13^{\sqrt{x+1.1}}, \quad x=3$(b) $h(x)=\left(2^{x}+2^{-x}\right)^{2 x}, \quad x=1.06$, Approximate the function at the value of $x$ to four decimal places. For the graph on the right, the base is a number between 0 and 1. Federal government receipts (in billions of dollars) for selected years are listed in the table.$$\begin{array}{|l|c|c|c|c|}\hline \text { Year } & 1910 & 1930 & 1950 & 1970 \\\hline \text { Receipts } & 0.7 & 4.1 & 39.4 & 192.8 \\\hline\end{array}$$$$\begin{array}{|l|l|l|l|}\hline \text { Year } & 1980 & 1990 & 2000 \\\hline \text { Receipts } & 517.1 & 1032.0 & 2025.2 \\\hline\end{array}$$(a) Let $x=0$ correspond to the year $1910 .$ Plot the data, together with the functions fand $g$. 1. When graphing with a calculator, we use the fact that the calculator can compute only common logarithms (base is [latex]10[/latex]), natural logarithms (base is [latex]e[/latex]) or binary logarithms (base is [latex]2[/latex]). In $1974,$ Johnny Miller won 8 tournaments on the PGA tour and accumulated 353,022 dollars in official season earnings. Suppose that for an initial dose of 10 milligrams, the amount $A(t)$ in the body $t$ hours later is given by $A(t)=10(0.8)^{t}$. Approximate the value of the house, to the nearest 1000 dollars, in the year 2016 . (b) As $x \rightarrow-\infty, f(x) \rightarrow$_____. Complete the statements for $f(x)=a^{x}+c$ with $a>1$. Answers may vary. (b) What percentage of the drug still in the body is eliminated each hour? (a) Estimate $y$ if $x=40 .$ (b) Estimate $x$ if $y=2$.$$y=(1.0525)^{x}$$, Use a graph to estimate the roots of the equation.$$1.4 x^{2}-2.2^{x}=1$$, Use a graph to estimate the roots of the equation.$$1.21^{3 x}+1.4^{-1.1 x}-2 x=0.5$$, Graph $f$ on the given interval. We first write the function as an equation as follows y = ex-3 2. 10 days? But before you take a look at the worked examples, I suggest that you review the suggested steps below first in order to have a good grasp of the general procedure. This section is about the inverse of the exponential function. Use the sliders below the graphs to change the values of b, the base of the logarithmic function y = log b x and its corresponding exponential function y = b x. (c) Use this curve to predict the cost of a 30 -second commercial in $2002 .$ Compare your answer to the actual value of 1,900,000 dollars. Find $k$. Approximate the function at the value of $x$ to four decimal places. If 10 grams of salt is added to a quantity of water, then the amount $q(t)$ that is undissolved after $t$ minutes is given by $q(t)=10\left(\frac{4}{5}\right)^{t} .$ Sketch a graph that shows the value $q(t)$ at any time from $t=0$ to $t=10$. The inverse of an exponential function is a logarithmic function. (a) $f(x)=2^{\sqrt{1-x}}, \quad x=0.5$(b) $h(x)=\frac{3^{-x}+5}{3^{x}-16}, \quad x=1.4$, Sketch the graph of the equation. The estimated number of new cases every 28 days is listed in the table. (b) Determine a curve in the form $y=a b^{x}$, where $x=0$ is the first year and $y$ is the cost that models the data. (a) Estimate the number alive after 5 years. (a) Determine whether $f$ is one-to-one. Glottochronology is a method of dating a language at a particular stage, based on the theory that over a long period of time linguistic changes take place at a fairly constant rate. (a) If the initial value of the item is $y_{0},$ show that the value after $n$ years of depreciation is $(1-a)^{n} y_{0}$. Graph, on the same coordinate plane, the line $y=1 / k$ and the logistic function with $k=\frac{1}{4}, a=\frac{1}{8},$ and $b=\frac{5}{8} .$ What is the significance of the value $1 / k ?$, If monthly payments $p$ are deposited in a savings account paying an annual interest rate $r,$ then the amount $A$ in the account after $n$ years is given by$$A=\frac{p\left(1+\frac{r}{12}\right)\left[\left(1+\frac{r}{12}\right)^{12 n}-1\right]}{\frac{r}{12}}$$ Graph $A$ for each value of $p$ and $r,$ and estimate $n$ for $A=100,000 \text{dollars}$.$$p=100, \quad r=0.05$$, If monthly payments $p$ are deposited in a savings account paying an annual interest rate $r,$ then the amount $A$ in the account after $n$ years is given by$$A=\frac{p\left(1+\frac{r}{12}\right)\left[\left(1+\frac{r}{12}\right)^{12 n}-1\right]}{\frac{r}{12}}$$ Graph $A$ for each value of $p$ and $r,$ and estimate $n$ for $A=100,000 \text{dollars}$.$$p=250, \quad r=0.09$$. One hundred elk, each 1 year old, are introduced into a game preserve. (c) Determine the date on which the number of new cases peaked. The declining balance method is an accounting method in which the amount of depreciation taken each year is a fixed percentage of the present value of the item. If a savings fund pays interest at a rate of $3\%$ per year compounded semiannually, how much money invested now will amount to 5000 dollars after 1 year? Inverse, Exponential, and Logarithmic Functions, Precalculus Functions and Graphs 12th - Earl W. Swokowski, Jeffrey A. Cole | All the textbook answers and step… William Farr correctly predicted when the number of new cases would peak. (1) $f(x)=0.786(1.094)^{x}$(2) $g(x)=0.503 x^{2}-27.3 x+149.2$(b) Determine whether the exponential or quadratic function better models the data. The taxpayer wishes to choose a depreciation rate such that the value of the item after $T$ years will equal the salvage value (see the figure). The Island of Manhattan was sold for 24 dollars in $1626 .$ How much would this amount have grown to by 2012 if it had been invested at $6 \%$ per year compounded quarterly? 12.5 days? Credit-card interest A certain department store requires its credit-card customers to pay interest on unpaid bills at the rate of $24 \%$ per year compounded monthly. For the graph on the left, the base is a number greater than 1. (GRAPH CAN'T COPY). This website uses cookies to ensure you get the best experience. Inverse, Exponential, and Logarithmic Functions. Examples, with detailed solutions, on how to find the inverse of exponential functions and also their domain and range. It is predicted that the number $N(t)$ still alive after $t$ years will be given by the equation $N(t)=1000(0.9)^{t} .$ Use the graph of $N$ to approximate when 500 trout will be alive. (b) Find the annual interest rate needed for Miller's winnings to be equivalent in value to Woods's winnings. Free functions inverse calculator - find functions inverse step-by-step. Automobile trade-in value If a certain make of automobile is purchased for $C$ dollars, its trade-in value $V(t)$ at the end of $t$ years is given by $V(t)=0.78 C(0.85)^{t-1} .$ If the original cost is 25,000 dollars, calculate, to the nearest dollar, the value after(a) 1 year(b) 4 years(c) 7 years, Real estate appreciation If the value of real estate increases at a rate of 4\% per year, after $t$ years the value $V$ of a house purchased for $P$ dollars is $V=P(1.04)^{t} .$ A graph for the value of a house purchased for 80,000 dollars in 1986 is shown in the figure.

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