Graph y = 5x and y = log5x on a sheet of paper using the same set of axes. Use the graph to describe the domain and range of each function. Then identify the y-intercept of each function and any asymptotes of each function. Explain also.

Answer:

1) For  

A)  Domain=

B) Range=

C) y-intercept = 0

D) Asymptote= No asymptote

2) For  

A)  Domain=Domain=  

B) Range=

C) y-intercept =  None

D) Vertical Asymptote:   x=0

Step-by-step explanation:

Given : and

Refer the graph attached.

1)  In equation (1)  

→The domain is the set of all possible values in which function is defined.  

y=5x is a linear polynomial defined on all real numbers.

Domain=

→Range is the set of all values that function takes.

It also include all real numbers.

Range=

→y-intercept- Value of y at the point where the line crosses the y axis.

put x=0 in equation y=5x we get, y=0

Therefore, y-intercept = 0 (We can see in the graph also)

→An asymptote is a line that a curve approaches, but never touches.

Asymptote= No asymptote

2) Now in equation (2)

Domain=  

because log function is not defined in negative.

Range=  

y-intercept - None

Vertical Asymptote:   x=0

1)

A)  Domain= (-∞, ∞) for all x

B) Range= (-∞, ∞) for all y

C) y-intercept = 0

D) Asymptote= No asymptote

2)

A)  Domain=(0, ∞) for all x > 0

B) Range= (-∞, ∞) for all y

C) y-intercept =  None

D) Vertical Asymptote:   x=0

Here, we have,

Function 1: y = 5x

Domain: The domain of this function is all real numbers because there are no restrictions on the values that x can take.

Range: The range of this function is also all real numbers because for every value of x, we can find a corresponding y value by multiplying it by 5.

Y-intercept: To find the y-intercept, we set x = 0 and solve for y. Substituting x = 0 into the equation, we get y = 5(0) = 0. Therefore, the y-intercept is (0, 0).

Asymptotes: There are no asymptotes in this linear function.

Function 2: y = log₅x

Domain: The domain of this function is all positive real numbers because the logarithm function is only defined for positive values of x.

Range: The range of this function is all real numbers because the logarithm function can produce any real number output.

Y-intercept: To find the y-intercept, we set x = 1 and solve for y. Substituting x = 1 into the equation, we get y = log₅(1) = 0. Therefore, the y-intercept is (0, 0).

Asymptotes: The logarithmic function has a vertical asymptote at x = 0 because the logarithm is undefined for x ≤ 0. Additionally, there is no horizontal asymptote.

When plotting these functions on the same set of axes, we will observe that the graph of y = 5x is a straight line passing through the origin (0, 0) with a slope of 5.

The graph of y = log₅x will appear as a curve that starts at the point (1, 0) and approaches the vertical asymptote x = 0 as x approaches zero.

The two graphs will intersect at the point (1, 0) because log₅1 = 0.

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