Chapter 4, Problem 1RCC

a.

To determine

Write an equation that defines the exponential function with base $a$ .

Answer to Problem 1RCC

  $\boxed{f\left(x\right)=a^x}$

Explanation of Solution

Given information:

Write an equation that defines the exponential function with base $a$ .

Calculation:

The exponential function with base $a$ algebraically written as,

  $f\left(x\right)=a^x$

  $where,a>0,a\ne 1$

Note that we have $a>0$ and number such that $\frac{1}{2}$ otherwise would not be the domain. We cannot take square root of negative number, we required $a\ne 1$ because the exponential function would be a constant function.

Hence, the solution is $\boxed{f\left(x\right)=a^x}$

b.

To determine

What is the domain of the function?

Answer to Problem 1RCC

  $\boxed{D=\boxed{x}}$

Explanation of Solution

Given information:

What is the domain of the function?

Calculation:

The domain of a function, D, is all the values which the function assigns values. The exponential function is defined for all the real values of x, hence:

  $\boxed{D=\boxed{x}}$

Hence, the solution is $\boxed{D=\boxed{x}}$

c.

To determine

What is the range of the function?

Answer to Problem 1RCC

  $\boxed{R=\left(0,\infty \right)}$

Explanation of Solution

Given information:

What is the range of the function?

Calculation:

The range of function, denoted $R$ , is all the values the function outputs. For any given base, a, the general shape of the graph looks like:

  Hence, the solution is $\boxed{R=\left(0,\infty \right)}$

d.

To determine

Sketch the graph for exponential function.

Answer to Problem 1RCC

Graph.

Explanation of Solution

Given information:

Sketch the graph for exponential function for each case, $\left(i\right)a>0$ $,\left(ii\right)0<a<1$

Calculation:

The graph for exponential function for $\left(i\right)a>0$

  

The graph for exponential function for $,\left(ii\right)0<a<1$

  

Hence, the solution is shown in graph.