# Transformations of functions questions and answers

Recent questions in Transformations of functions
Transformations of functions

### Find the limit (if it exists) and discuss the continuity of the function. $$\lim_{(x,y,z) \rightarrow (-3,1,2)}\frac{\ln z}{xy-z}$$

Transformations of functions

### Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. $$\displaystyle{g{{\left({x}\right)}}}={2}\sqrt{{{3}}}-{x},{\left(-\infty,{3}\right]}$$

Transformations of functions

### Find the limit and discuss the continuity of the function. $$\displaystyle\lim_{{{x},{y}}}\rightarrow{\left({\frac{{\pi}}{{{4}}}},{2}\right)}{y}{\cos{{x}}}{y}$$

Transformations of functions

### Describe the transformations that must be applied to y=x^2 to create the graph of each of the following functions. a) $$\displaystyle{y}=\frac{{1}}{{4}}{\left({x}-{3}\right)}^{{2}}+{9}$$ b) $$\displaystyle{y}={\left({\left(\frac{{1}}{{2}}\right)}{x}\right)}^{{2}}-{7}$$

Transformations of functions

### Sketch the graph of the function f(x)=[x]]+[∣−x] $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{l}\right\rbrace}{\left\lbrace\ \text{ (a) Evaluate }\ {f{{\left({1}\right)}}},{f{{\left({0}\right)}}},{f}{\left({\frac{{{1}}}{{{2}}}}\right)},\ \text{ and }\ {f{{\left(-{2.7}\right)}}}\right\rbrace}\backslash{\left\lbrace\ \text{ (b) Evaluate the limits }\ \lim_{{{x}\rightarrow{1}^{{-}}}}{f{{\left({x}\right)}}},\lim_{{{x}\rightarrow{1}^{{+}}}}{f{{\left({x}\right)}}},\ \text{ and }\ \lim_{{{x}\rightarrow{1}&#{x}{2}{F}.{2}}}{f{{\left({x}\right)}}}\right\rbrace}\backslash{\left\lbrace\ \text{ (c) Discuss the continuity of the function. }\ \right\rbrace}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$

Transformations of functions

### Describe the transformations that were applied to $$y=x^3$$to create each of the following functions. a) $$\displaystyle{y}={12}{\left({x}−{9}\right)}^{{3}}−{7}$$ b) $$\displaystyle{y}={\left(\frac{{7}}{{8}}{\left({x}+{1}\right)}\right)}{3}+{3}$$ c) $$\displaystyle{y}=−{2}{\left({x}−{6}\right)}^{{3}}−{8}$$ d) $$y = (x + 9) (x + 9) (x + 9)$$ e) $$\displaystyle{y}=−{2}{\left(−{3}{\left({x}−{4}\right)}\right)}^{{3}}−{5}$$ f) $$\displaystyle{y}={\left(\frac{{3}}{{4}}{\left({x}−{10}\right)}\right)}^{{3}}$$

Transformations of functions

### g is related to one of the parent functions described in Section 1.6. Describe the sequence of transformations from f to g. g(x) = (x + 3)^3 - 10

Transformations of functions

### Find the limit and discuss the continuity of the function. $$\displaystyle\lim_{{{x},{y}}}\rightarrow{\left({2}\pi,{4}\right)}{\sin{{\frac{{{x}}}{{{y}}}}}}$$

Transformations of functions

### Describe the transformations that must be applied to the parent function to obtain each of the following functions. a) $$\displaystyle{f{{\left({x}\right)}}}=-{3}{\log{{\left({2}{x}\right)}}}$$ b) $$\displaystyle{f{{\left({x}\right)}}}={\log{{\left({x}-{5}\right)}}}+{2}$$ c) $$\displaystyle{f{{\left({x}\right)}}}={\left(\frac{{1}}{{2}}\right)}{\log{{5}}}{x}$$ d) $$\displaystyle{f{{\left({x}\right)}}}={\log{{\left(−{\left(\frac{{1}}{{3}}\right)}{x}\right)}}}−{3}$$

Transformations of functions

### Discuss the continuity of the function and evaluate the limit of f(x, y) (if it exists) as $$\displaystyle{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}.{f{{\left({x},{y}\right)}}}={1}-{\frac{{{\cos{{\left({x}^{{{2}}}+{y}^{{{2}}}\right)}}}}}{{{x}^{{{2}}}+{y}^{{{2}}}}}}$$

Transformations of functions

### Begin by graphing $$\displaystyle{f{{\left({x}\right)}}}={2}^{{{x}}}$$ Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$\displaystyle{g{{\left({x}\right)}}}={2}\cdot{2}^{{{x}}}$$

Transformations of functions

### For each of the following functions f (x) and g(x), express g(x) in the form $$\displaystyle{a}:{f{{\left({x}+{b}\right)}}}+{c}$$ for some values a,b and c, and hence describe a sequence of horizontal and vertical transformations which map $$\displaystyle{f{{\left({x}\right)}}}\ \to\ {g{{\left({x}\right)}}}.{\left({a}\right)}{\left({i}\right)}$$ $$\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}},{g{{\left({x}\right)}}}={2}{x}^{{{2}}}+{4}{x}$$ $$\displaystyle{\left({i}{i}\right)}{f{{\left({x}\right)}}}={x}^{{{2}}},{g{{\left({x}\right)}}}={3}{x}^{{{2}}}-{24}{x}+{8}$$ $$\displaystyle{\left({b}\right)}{\left({i}\right)}{f{{\left({x}\right)}}}={x}^{{{2}}}+{3},{g{{\left({x}\right)}}}={x}^{{{2}}}-{6}{x}+{8}$$ $$\displaystyle{\left({i}{i}\right)}{f{{\left({x}\right)}}}={x}^{{{2}}}-{2},{g{{\left({x}\right)}}}={2}+{8}{x}-{4}{x}^{{{2}}}$$

Transformations of functions

### For $$\displaystyle{y}={{\log}_{{{3}}}{\left({x}+{2}\right)}}$$ a. Use transformations of the graphs of $$\displaystyle{y}={{\log}_{{{2}}}{x}}$$ and $$\displaystyle{y}={{\log}_{{{3}}}{x}}$$ to graph the given functions. b. Write the domain and range in interval notation. c. Write an equation of the asymptote.

Transformations of functions

### Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m+mf. Continuing in this fashion, the amount of medication in your blood just after your nth does is $$\displaystyle{A}_{{{n}}}={m}+{m}{f}+\ldots+{m}{f}^{{{n}-{1}}}$$. For the given values off and m, calculate $$\displaystyle{A}_{{{5}}},{A}_{{{10}}},{A}_{{{30}}}$$, and $$\displaystyle\lim_{{{n}\rightarrow\infty}}{A}_{{{n}}}$$. Interpret the meaning of the limit $$\displaystyle\lim_{{{n}\rightarrow\infty}}{A}_{{{n}}}$$. f=0.25,m=200mg.

Transformations of functions

### Begin by graphing $$\displaystyle{f{{\left({x}\right)}}}={2}^{{{x}}}$$ Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$\displaystyle{h}{\left({x}\right)}={2}^{{{x}+{1}}}-{1}$$

Transformations of functions

### For $$\displaystyle{y}={2}+{{\log}_{{{3}}}{x}}$$. a) Use transformations of the graphs of $$\displaystyle{y}={{\log}_{{{2}}}{x}}\ {\quad\text{and}\quad}\ {y}={{\log}_{{{3}}}{x}}$$ o graph the given functions. b) Write the domain and range in interval notation. c) Write an equation of the asymptote.

Transformations of functions

### Sketch a graph of the function. Use transformations of functions whenever possible. $$\displaystyle{f{{\left({x}\right)}}}=-{\frac{{{1}}}{{{x}^{{{2}}}}}}$$

Transformations of functions

### In your earlier work in algebra, you learned how to recognize linear, exponential, and quadratic functions by the form of their symbolic rules. Geometric transformations also can be recognized by their symbolic rules. What transformation is defined by each of the following coordinate rules? $$\displaystyle{a}.{\left({x},{y}\right)}\rightarrow{\left({5}{y},{5}{x}\right)}\ {b}.{\left({x},{y}\right)}\rightarrow{\left(-{\frac{{{1}}}{{{2}}}}{x},-{\frac{{{1}}}{{{2}}}}{y}\right)}\ {c}.{\left({x},{y}\right)}\rightarrow{\left({4}{x}-{12},{4}{y}+{8}\right)}$$

Transformations of functions