Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.
a. g(1)

Verified step by step guidance

1. Step 1: Understand the problem. We are given a function g(x) = f(x^2) and need to find g(1). We know that f is differentiable and passes through the point (1, 4), and the tangent line to f at (1, 4) is y = 3x + 1.
2. Step 2: Evaluate g(1). Since g(x) = f(x^2), we substitute x = 1 into g(x) to get g(1) = f(1^2) = f(1).
3. Step 3: Use the information about f. We know that the graph of f passes through the point (1, 4), which means f(1) = 4.
4. Step 4: Conclude the evaluation. Since f(1) = 4, we have g(1) = f(1) = 4.
5. Step 5: Summarize the result. Therefore, g(1) is equal to 4.

Key Concepts

Differentiable Functions

A differentiable function is one that has a derivative at every point in its domain. This means that the function is smooth and continuous, allowing for the calculation of slopes of tangent lines at any point. The existence of a derivative indicates that the function can be locally approximated by a linear function, which is essential for understanding tangent lines.

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. The equation of the tangent line can be derived using the point-slope form, which incorporates the derivative of the function at that point. In this case, the tangent line to f at (1, 4) is given as y = 3x + 1, indicating that the slope at that point is 3.

Composition of Functions

The composition of functions involves combining two functions where the output of one function becomes the input of another. In this problem, g(x) = f(x²) represents a composition where the input x is squared before being passed to function f. Understanding how to evaluate g(1) requires substituting 1 into the composition and then determining the value of f at the resulting input.